PMATH 867: Geometric Group Theory

Estimated study time: 1 hr 58 min

Table of contents

These notes synthesize material from M. Bridson and A. Haefliger’s Metric Spaces of Non-Positive Curvature, C. Drutu and M. Kapovich’s Geometric Group Theory, C. Löh’s Geometric Group Theory: An Introduction, and P. de la Harpe’s Topics in Geometric Group Theory, enriched with material from B. Bowditch’s course notes and M. Bestvina’s lecture notes.

Chapter 1: Groups as Geometric Objects

The central insight of geometric group theory is that finitely generated groups are themselves geometric objects. Rather than studying groups purely through their algebraic structure — subgroups, quotients, homomorphisms — we equip them with metrics arising from their generators and study them through the lens of large-scale geometry. This perspective, crystallized by Gromov in his landmark 1987 paper on hyperbolic groups, transformed the landscape of combinatorial and geometric group theory by revealing that coarse geometric properties of groups carry deep algebraic information.

1.1 Cayley Graphs and Word Metrics

To view a group as a geometric object, we need to assign it a metric space structure. The classical construction achieving this is the Cayley graph.

Let \(G\) be a group and \(S \subseteq G\) a generating set with \(S = S^{-1}\) (i.e., \(S\) is symmetric). The Cayley graph \(\mathrm{Cay}(G, S)\) is the graph whose vertex set is \(G\) and whose edge set consists of pairs \(\{g, gs\}\) for all \(g \in G\) and \(s \in S\).

The Cayley graph is a connected graph (since \(S\) generates \(G\)), and it admits a natural left action of \(G\) by graph automorphisms: for each \(h \in G\), the map \(g \mapsto hg\) sends edges to edges. This action is free and transitive on vertices, so the Cayley graph is vertex-transitive.

Example 1.1. The Cayley graph \(\mathrm{Cay}(\mathbb{Z}, \{1, -1\})\) is the integer line — a bi-infinite path. The Cayley graph \(\mathrm{Cay}(\mathbb{Z}^2, \{(\pm 1, 0), (0, \pm 1)\})\) is the standard square grid in \(\mathbb{R}^2\). The Cayley graph \(\mathrm{Cay}(F_2, \{a^{\pm 1}, b^{\pm 1}\})\) of the free group on two generators is the infinite 4-regular tree.

We metrize the Cayley graph by declaring each edge to have length one and taking the induced path metric. Restricting this metric to the vertex set gives us the word metric on the group.

Let \(G\) be a group generated by a symmetric set \(S\). The word metric \(d_S : G \times G \to \mathbb{Z}_{\ge 0}\) is defined by \[ d_S(g, h) = \min \{ n \ge 0 : h = g s_1 s_2 \cdots s_n, \; s_i \in S \}. \] Equivalently, \(d_S(g, h)\) is the length of the shortest path from \(g\) to \(h\) in \(\mathrm{Cay}(G, S)\). The word length (or word norm) of an element \(g \in G\) is \(|g|_S = d_S(e, g)\).

The word metric is left-invariant: \(d_S(hg, hg') = d_S(g, g')\) for all \(h, g, g' \in G\). This follows immediately from the fact that \(h\) acts by isometries on the Cayley graph.

A natural concern is that the Cayley graph and word metric depend on the choice of generating set. Different generating sets produce genuinely different metric spaces. However, as we shall see, the large-scale geometry remains the same — and this is made precise by the notion of quasi-isometry.

1.2 Quasi-Isometries

The appropriate notion of equivalence for the large-scale geometry of metric spaces is not isometry but something much coarser: quasi-isometry. This is the key notion that allows us to speak of geometric properties of a group independent of the choice of generators.

Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. A map \(f: X \to Y\) is a quasi-isometric embedding if there exist constants \(L \ge 1\) and \(C \ge 0\) such that for all \(x, x' \in X\), \[ \frac{1}{L} d_X(x, x') - C \le d_Y(f(x), f(x')) \le L \cdot d_X(x, x') + C. \] If, in addition, there exists \(D \ge 0\) such that every point of \(Y\) lies within distance \(D\) of the image of \(f\) (i.e., \(f\) is coarsely surjective or a quasi-surjection), then \(f\) is a quasi-isometry. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them.

Quasi-isometry is an equivalence relation on metric spaces (given appropriate conditions on the spaces). A quasi-inverse of a quasi-isometry \(f: X \to Y\) is a quasi-isometry \(\bar{f}: Y \to X\) such that \(d_X(\bar{f} \circ f(x), x)\) and \(d_Y(f \circ \bar{f}(y), y)\) are uniformly bounded.

Proposition 1.2. If \(S\) and \(S'\) are two finite symmetric generating sets for a group \(G\), then the identity map \(\mathrm{id}: (G, d_S) \to (G, d_{S'})\) is a quasi-isometry.

Since \(S\) generates \(G\), each element of \(S'\) can be written as a product of elements of \(S\). Let \(L = \max_{s' \in S'} |s'|_S\). Then for any \(g \in G\), \(|g|_{S'} \le |g|_S\) (up to the factor \(L\)), and similarly in the other direction. More precisely, \(|g|_{S'} \le L \cdot |g|_S\) and \(|g|_S \le L' \cdot |g|_{S'}\) where \(L' = \max_{s \in S} |s|_{S'}\). Setting \(M = \max(L, L')\), we have \[ \frac{1}{M} d_S(g, h) \le d_{S'}(g, h) \le M \cdot d_S(g, h) \] for all \(g, h \in G\), which shows the identity is a bilipschitz equivalence (quasi-isometry with \(C = 0\)). \(\blacksquare\)

This proposition justifies speaking of “the large-scale geometry of \(G\)” without specifying a generating set. Two groups are quasi-isometric if and only if their Cayley graphs (for any choice of finite generating sets) are quasi-isometric as metric spaces.

Example 1.3. The inclusion \(\mathbb{Z} \hookrightarrow \mathbb{R}\) is a quasi-isometry. Indeed, it is an isometric embedding (with \(L = 1, C = 0\)) and every real number is within distance \(1/2\) of an integer. More generally, any finitely generated group \(G\) is quasi-isometric to its Cayley graph \(\mathrm{Cay}(G,S)\) (viewed as a metric graph with edges of length one), since the inclusion of the vertex set is a quasi-isometry.

Example 1.4. Any two finite groups are quasi-isometric to each other (and to a point). Any two finitely generated groups that differ by a finite normal subgroup are quasi-isometric: if \(N \trianglelefteq G\) is finite, then the quotient map \(G \to G/N\) is a quasi-isometry.

1.3 The Milnor-Schwarz Lemma

The Milnor-Schwarz lemma (also called the Schwarz-Milnor lemma or the fundamental observation of geometric group theory) is the bridge between the geometry of spaces and the geometry of groups acting on them. It was proved independently by Milnor (1968) and Schwarz (1955).

Theorem 1.5 (Milnor-Schwarz Lemma). Let \(X\) be a proper geodesic metric space, and let \(G\) be a group acting properly discontinuously and cocompactly by isometries on \(X\). Then \(G\) is finitely generated and, for any \(x_0 \in X\), the orbit map \[ G \to X, \quad g \mapsto g \cdot x_0 \] is a quasi-isometry (where \(G\) is equipped with a word metric for some finite generating set).

Let \(K \subseteq X\) be a compact set such that \(G \cdot K = X\) (cocompactness). We may assume \(x_0 \in K\). Since \(X\) is proper and the action is by isometries, we can choose \(R > 0\) large enough so that \(K \subseteq B(x_0, R)\). Let \[ S = \{ g \in G \setminus \{e\} : g \cdot B(x_0, R) \cap B(x_0, R) \neq \emptyset \}. \] Since the action is properly discontinuous and \(\overline{B(x_0, 2R)}\) is compact (properness), the set \(S\) is finite. We claim \(S\) generates \(G\).

Indeed, let \(g \in G\). Since \(X\) is geodesic, let \(\gamma\) be a geodesic from \(x_0\) to \(g \cdot x_0\). Choose points \(x_0 = y_0, y_1, \ldots, y_n = g \cdot x_0\) along \(\gamma\) with \(d(y_i, y_{i+1}) \le R\). By cocompactness, for each \(y_i\) there exists \(g_i \in G\) with \(d(g_i \cdot x_0, y_i) \le R\). Setting \(g_0 = e\) and \(g_n = g\), we see that \(d(g_i \cdot x_0, g_{i+1} \cdot x_0) \le 3R\), so \(g_i^{-1} g_{i+1} \in S \cup \{e\}\) (after possibly enlarging \(R\)). Thus \(g = (g_0^{-1} g_1)(g_1^{-1} g_2) \cdots (g_{n-1}^{-1} g_n)\) is a product of at most \(n\) elements of \(S \cup S^{-1}\), showing \(S\) generates and \(|g|_S \le n \le d(x_0, g \cdot x_0)/R + 1\).

\[ d(x_0, g \cdot x_0) \le \sum_{i=1}^n d(s_1 \cdots s_{i-1} \cdot x_0, s_1 \cdots s_i \cdot x_0) = \sum_{i=1}^n d(x_0, s_i \cdot x_0) \le 3Rn. \]

Combining both bounds shows the orbit map \(g \mapsto g \cdot x_0\) is a quasi-isometric embedding. Coarse surjectivity follows from cocompactness: every point of \(X\) lies within distance \(R\) of the orbit \(G \cdot x_0\). \(\blacksquare\)

The Milnor-Schwarz lemma has far-reaching consequences. It tells us that the quasi-isometry class of a group captures not just algebraic information, but also the large-scale geometry of any space on which the group acts nicely.

Corollary 1.6. The fundamental group of a closed Riemannian manifold \(M\) is quasi-isometric to the universal cover \(\widetilde{M}\) (equipped with the lifted Riemannian metric).

Example 1.7. The fundamental group of a closed hyperbolic surface \(\Sigma_g\) (genus \(g \ge 2\)) is quasi-isometric to the hyperbolic plane \(\mathbb{H}^2\). The group \(\mathbb{Z}^n\) is quasi-isometric to \(\mathbb{R}^n\) (acting by translations). The free group \(F_k\) on \(k \ge 2\) generators is quasi-isometric to a regular tree of valence \(2k\).

1.4 Quasi-Isometry Invariants

A property or quantity associated to metric spaces (or groups) is a quasi-isometry invariant if it is preserved under quasi-isometries. Identifying quasi-isometry invariants is one of the central goals of geometric group theory, as they allow us to distinguish groups up to quasi-isometry and to transfer geometric information between a group and the spaces on which it acts.

Proposition 1.8. The following are quasi-isometry invariants of finitely generated groups:

Being finitely presented.
Being virtually nilpotent, virtually abelian, virtually free, or virtually cyclic.
The number of ends.
Hyperbolicity (in the sense of Gromov).
The growth type (polynomial, exponential, or intermediate).
Amenability.
The isoperimetric function (Dehn function), up to the natural equivalence relation.

We will encounter most of these invariants in subsequent chapters. For now, let us focus on the growth of groups.

1.5 Growth of Groups

The growth function of a finitely generated group measures how the sizes of metric balls in the Cayley graph increase with radius. This notion was introduced by Schwarz (1955) and studied systematically by Milnor (1968) and Wolf (1968).

Let \(G\) be a group generated by a finite symmetric set \(S\). The growth function of \(G\) with respect to \(S\) is the function \[ \beta_S(n) = |B_S(e, n)| = |\{ g \in G : |g|_S \le n \}|. \]

Although \(\beta_S\) depends on the generating set, its asymptotic behavior does not. Two functions \(f, g: \mathbb{N} \to \mathbb{R}_+\) have the same growth type if there exists \(C > 0\) such that \(f(n) \le g(Cn)\) and \(g(n) \le f(Cn)\) for all \(n\). This is an equivalence relation, and the growth type of \(\beta_S\) is independent of \(S\) — indeed, it is a quasi-isometry invariant.

A finitely generated group \(G\) has:

Polynomial growth (of degree \(d\)) if \(\beta_S(n) \preceq n^d\) for some \(d\) (i.e., there exists \(C > 0\) with \(\beta_S(n) \le C n^d\) for all \(n\)).
Exponential growth if \(\beta_S(n) \succeq e^n\) (i.e., there exist \(c, a > 0\) with \(\beta_S(n) \ge c \cdot a^n\) for all \(n\), where \(a > 1\)).
Intermediate growth if it has neither polynomial nor exponential growth.

Example 1.9. The group \(\mathbb{Z}^d\) has polynomial growth of degree \(d\), as \(\beta_S(n) \sim c \cdot n^d\) for the standard generators. The free group \(F_k\) (\(k \ge 2\)) has exponential growth: with respect to the standard generators, \(\beta_S(n) = 1 + 2k \cdot \frac{(2k-1)^n - 1}{2k - 2}\). More generally, any group containing a non-abelian free subgroup has exponential growth.

The classification of growth types is intimately related to the algebraic structure of the group. Milnor and Wolf proved in the late 1960s:

Theorem 1.10 (Milnor-Wolf). A finitely generated solvable group has polynomial growth if and only if it is virtually nilpotent. Otherwise, it has exponential growth.

In particular, solvable groups have no intermediate growth. This result was dramatically generalized by Gromov:

Theorem 1.11 (Gromov, 1981). A finitely generated group has polynomial growth if and only if it is virtually nilpotent.

Gromov’s proof is a tour de force, using the Milnor-Schwarz lemma to pass from the group to an asymptotic cone, showing the asymptotic cone is locally compact and finite-dimensional, appealing to the solution of Hilbert’s fifth problem (Montgomery-Zippin) to conclude the asymptotic cone is a Lie group, and then using structural results for Lie groups. A simpler proof was later given by Kleiner (2010) using harmonic functions on groups.

Remark 1.12. The converse direction — that virtually nilpotent groups have polynomial growth — was known earlier and follows from the Bass-Guivarc'h formula: if \(G\) is a finitely generated nilpotent group with lower central series \(G = G_1 \supsetneq G_2 \supsetneq \cdots \supsetneq G_{c+1} = \{e\}\), then \(\beta_S(n) \sim n^d\) where \[ d = \sum_{i=1}^{c} i \cdot \mathrm{rank}(G_i / G_{i+1}). \] In particular, the Heisenberg group \(H_3(\mathbb{Z})\) has polynomial growth of degree 4.

An outstanding question in the 1960s and 1970s was whether intermediate growth actually occurs. This was answered affirmatively by Grigorchuk:

Theorem 1.13 (Grigorchuk, 1984). The Grigorchuk group \(\Gamma\) is a finitely generated infinite torsion group of intermediate growth. More precisely, \[ e^{n^{0.504}} \preceq \beta_S(n) \preceq e^{n^{0.767}}. \]

The Grigorchuk group is defined as a group of automorphisms of the infinite rooted binary tree. Its existence settled several open problems simultaneously: it provided the first example of intermediate growth, a finitely generated infinite torsion group (answering the Burnside problem in the finitely generated setting), and an amenable group that is not elementary amenable.

Chapter 2: Free Groups and Presentations

Free groups are the most fundamental building blocks in combinatorial group theory. They play a role analogous to that of vector spaces in linear algebra or polynomial rings in commutative algebra: every group is a quotient of a free group, and the study of how groups are presented as quotients of free groups leads directly to the word problem and its geometric manifestations through van Kampen diagrams and Dehn functions.

2.1 Free Groups

Let \(X\) be a set. The free group \(F(X)\) on \(X\) is a group \(F(X)\) together with a map \(\iota: X \to F(X)\) satisfying the following universal property: for every group \(G\) and every map \(\varphi: X \to G\), there exists a unique homomorphism \(\Phi: F(X) \to G\) such that \(\Phi \circ \iota = \varphi\).

The existence of free groups can be established via reduced words. Let \(X^{-1} = \{x^{-1} : x \in X\}\) be a formal set of inverses, and set \(X^{\pm 1} = X \cup X^{-1}\). A word \(w = a_1 a_2 \cdots a_n\) in the alphabet \(X^{\pm 1}\) is reduced if it contains no subword of the form \(x x^{-1}\) or \(x^{-1} x\). The set of reduced words forms a group under concatenation followed by free reduction, and this group satisfies the universal property above.

Proposition 2.1. The free group \(F(X)\) is well-defined (up to unique isomorphism) for any set \(X\). If \(|X| = n < \infty\), we write \(F_n = F(X)\) and call it the free group of rank \(n\).

The algebraic structure of free groups reflects their geometric nature: \(F_n\) acts freely and cocompactly on a tree (its Cayley graph), and this tree structure is responsible for many of the remarkable properties of free groups.

Theorem 2.2 (Nielsen-Schreier). Every subgroup of a free group is free.

The proof uses the fact that \(F_n\) acts freely on a tree \(T\) (its Cayley graph). If \(H \le F_n\), then \(H\) also acts freely on \(T\). The quotient \(H \backslash T\) is a graph, and by the general theory of groups acting on trees (a precursor to Bass-Serre theory), a group acting freely on a tree is free. More precisely, \(H\) is the fundamental group of the graph \(H \backslash T\), which is free of rank \(1 - \chi(H \backslash T)\) where \(\chi\) denotes the Euler characteristic. \(\blacksquare\)

Corollary 2.3 (Schreier Index Formula). If \(H\) is a subgroup of index \(k\) in \(F_n\), then \(H\) is free of rank \(k(n-1) + 1\).

This formula has the initially surprising consequence that subgroups of free groups can have higher rank than the ambient group: for instance, the commutator subgroup \([F_2, F_2]\) has infinite rank.

2.2 Group Presentations

A group presentation is an expression \(\langle X \mid R \rangle\) where \(X\) is a set (of generators) and \(R\) is a set of words in \(X^{\pm 1}\) (the relators). The group defined by this presentation is \[ G = \langle X \mid R \rangle = F(X) / \langle\!\langle R \rangle\!\rangle \] where \(\langle\!\langle R \rangle\!\rangle\) denotes the normal closure of \(R\) in \(F(X)\). A group is finitely presented if it admits a presentation with both \(X\) and \(R\) finite.

Example 2.4. Some familiar presentations:

\(\mathbb{Z} = \langle a \mid \rangle\), the infinite cyclic group.
\(\mathbb{Z}/n\mathbb{Z} = \langle a \mid a^n \rangle\).
\(\mathbb{Z}^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle\).
The fundamental group of a closed orientable surface of genus \(g\): \(\pi_1(\Sigma_g) = \langle a_1, b_1, \ldots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle\).

Theorem 2.5 (Tietze, 1908). Two finite presentations define isomorphic groups if and only if one can be obtained from the other by a finite sequence of Tietze transformations:

Adding a relator that is a consequence of the existing relators.
Removing a relator that is a consequence of the remaining relators.
Adding a new generator \(y\) and a relator \(y = w\) where \(w\) is a word in the existing generators.
Removing a generator \(y\) and a relator \(y = w\), substituting \(w\) for \(y\) in all other relators.

2.3 Dehn’s Problems

In 1911, Max Dehn posed three fundamental decision problems for finitely presented groups:

Let \(G = \langle X \mid R \rangle\) be a finitely presented group.

The word problem: Given a word \(w\) in \(X^{\pm 1}\), determine whether \(w =_G e\).
The conjugacy problem: Given words \(u, v\) in \(X^{\pm 1}\), determine whether \(u\) and \(v\) are conjugate in \(G\).
The isomorphism problem: Given two finite presentations, determine whether they define isomorphic groups.

These problems are of central importance in both algebra and topology. The word problem for \(\pi_1(M)\) is equivalent to determining whether a closed loop in \(M\) is null-homotopic; the isomorphism problem for fundamental groups of manifolds is closely related to the homeomorphism problem.

A landmark result of Novikov (1955) and Boone (1959) shows that the word problem is in general undecidable:

Theorem 2.6 (Novikov-Boone). There exists a finitely presented group whose word problem is algorithmically unsolvable.

Similarly, Adyan and Rabin showed that the isomorphism problem is unsolvable in general. However, for many geometrically defined classes of groups (hyperbolic groups, CAT(0) groups, automatic groups), the word problem is solvable, and geometric group theory provides the tools to prove this.

2.4 Van Kampen Diagrams

Van Kampen diagrams are the bridge between the combinatorial word problem and geometry. They provide a geometric way to certify that a word represents the identity in a group.

Let \(G = \langle X \mid R \rangle\) be a finitely presented group. A van Kampen diagram (or disk diagram) over this presentation is a finite, planar, connected, simply connected 2-complex \(D\) together with a labeling of its oriented edges by elements of \(X^{\pm 1}\) such that:

For each 2-cell (face) \(f\) of \(D\), the boundary label of \(f\) (read starting from some vertex and going around the boundary) is a cyclic permutation of a word in \(R \cup R^{-1}\).
Reading the boundary label of the outer boundary of \(D\) gives a word \(w\).

We say that \(D\) is a van Kampen diagram for the word \(w\).

Theorem 2.7 (van Kampen's Lemma). A word \(w\) in \(X^{\pm 1}\) represents the identity in \(G = \langle X \mid R \rangle\) if and only if there exists a van Kampen diagram for \(w\).

The number of 2-cells in a van Kampen diagram for \(w\) is called its area. The minimal area over all diagrams for \(w\) is denoted \(\mathrm{Area}(w)\), and it equals the minimal number of conjugates of relators needed to express \(w\) as a product in \(F(X)\).

2.5 Dehn Functions and Isoperimetric Inequalities

Let \(G = \langle X \mid R \rangle\) be a finitely presented group. The Dehn function (or isoperimetric function) of this presentation is \[ \delta(n) = \max \{ \mathrm{Area}(w) : w =_G e, \; |w| \le n \}. \]

The Dehn function measures the worst-case difficulty of the word problem: to certify that a word of length \(n\) represents the identity, one needs at most \(\delta(n)\) applications of relations. Two Dehn functions \(\delta_1, \delta_2\) are equivalent (written \(\delta_1 \simeq \delta_2\)) if each is bounded above by a linear function of the other (with an appropriate rescaling of the argument). The equivalence class of the Dehn function is independent of the presentation and is a quasi-isometry invariant.

Example 2.8. The Dehn function of \(\mathbb{Z}^2 = \langle a, b \mid [a,b] \rangle\) is quadratic: \(\delta(n) \simeq n^2\). The Dehn function of the free group \(F_k\) is linear: \(\delta(n) = 0\) since the free group has no non-trivial relations. The Dehn function of the Baumslag-Solitar group \(\mathrm{BS}(1,2) = \langle a, b \mid bab^{-1} = a^2 \rangle\) is exponential.

The terminology “isoperimetric inequality” comes from the analogy with Riemannian geometry: the Dehn function bounds the area of a filling disk in terms of the length of the boundary curve, just as an isoperimetric inequality in a Riemannian manifold bounds the area of a minimal surface in terms of the length of its boundary.

Theorem 2.9. A finitely presented group has a solvable word problem if and only if its Dehn function is bounded above by a recursive function.

2.6 Small Cancellation Theory

Small cancellation theory, developed by Tartakovskii, Greendlinger, Lyndon, and Schupp in the 1960s, provides geometric conditions on presentations that guarantee good algorithmic and structural properties. The basic idea is that if the relators in a presentation have little overlap (small cancellation), then van Kampen diagrams have special combinatorial properties that can be exploited.

Let \(\langle X \mid R \rangle\) be a group presentation where \(R\) is symmetrized (closed under cyclic permutations and inversion). A piece is a common prefix of two distinct elements of \(R\). The presentation satisfies:

The \(C'(\lambda)\) condition if every piece \(p\) that is a subword of a relator \(r \in R\) satisfies \(|p| < \lambda |r|\).
The \(T(q)\) condition if for any \(3 \le k < q\) and any relators \(r_1, \ldots, r_k \in R\) with \(r_i \neq r_{i+1}^{-1}\), at least one of the consecutive products \(r_i r_{i+1}\) is freely reduced as written.

The classical \(C'(1/6)\) condition (or more generally \(C'(1/6)\)-\(T(3)\)) is particularly powerful:

Theorem 2.10 (Greendlinger's Lemma). Let \(G = \langle X \mid R \rangle\) satisfy \(C'(1/6)\). If \(w =_G e\) and \(w\) is a non-empty freely reduced word, then \(w\) contains a subword \(s\) that is a subword of some relator \(r \in R\) with \(|s| > |r|/2\). In particular:

The group \(G\) is torsion-free (if all relators have length \(\ge 2\)).
The Dehn function of \(G\) is linear.
The word and conjugacy problems for \(G\) are solvable.

Greendlinger’s lemma implies that \(C'(1/6)\) groups satisfy a linear isoperimetric inequality. Moreover, it provides Dehn’s algorithm for the word problem: given a word \(w\), search for a subword \(s\) of \(w\) that is more than half of some relator; if found, replace \(s\) by the complementary shorter half of the relator, reducing the length of \(w\). Repeat. If no such subword exists and \(w\) is non-empty, then \(w \neq_G e\). This algorithm terminates in at most \(|w|\) steps.

As we shall see in Chapter 4, \(C'(1/6)\) groups are hyperbolic, and Dehn’s algorithm generalizes to all hyperbolic groups.

Chapter 3: Bass-Serre Theory

Bass-Serre theory, developed by Jean-Pierre Serre in his 1977 book Arbres, amalgames, \(\mathrm{SL}_2\) (with algebraic formalization by Hyman Bass), provides a systematic framework for studying groups acting on trees. The theory gives a complete dictionary between group actions on trees and algebraic decompositions of groups as fundamental groups of graphs of groups. It unifies and generalizes the classical constructions of free products with amalgamation and HNN extensions.

3.1 Group Actions on Trees

A tree is a connected graph with no cycles. Equivalently, a connected graph \(T\) is a tree if and only if any two vertices are connected by a unique reduced edge path. Trees are the simplest examples of CAT(0) spaces (as we will see in Chapter 5), and they play a distinguished role in geometric group theory.

An action of a group \(G\) on a tree \(T\) is without inversions if no element of \(G\) maps an edge to itself with reversed orientation. Given an action without inversions, we say:

The action is trivial if \(G\) fixes a vertex.
An element \(g \in G\) is elliptic if it fixes a vertex.
An element \(g \in G\) is hyperbolic if it fixes no vertex; in this case, \(g\) translates along a unique bi-infinite geodesic line \(\ell_g \subseteq T\) (its axis).

Proposition 3.1. Let \(G\) act on a tree \(T\) without inversions. Then every element of \(G\) is either elliptic or hyperbolic. An element \(g\) is hyperbolic if and only if it has infinite order and its minimum displacement \(\min_{v} d(v, gv)\) is positive.

3.2 Free Products with Amalgamation

Let \(A\) and \(B\) be groups with a common subgroup \(C\) (more precisely, let \(\varphi_A: C \hookrightarrow A\) and \(\varphi_B: C \hookrightarrow B\) be injective homomorphisms). The amalgamated free product \(A *_C B\) is the quotient of the free product \(A * B\) by the normal subgroup generated by \(\{\varphi_A(c)^{-1} \varphi_B(c) : c \in C\}\). Equivalently, it is the pushout in the category of groups: \[ A *_C B = \langle A, B \mid \varphi_A(c) = \varphi_B(c) \text{ for all } c \in C \rangle. \]

Example 3.2. The free product \(A * B\) is the special case \(C = \{e\}\). The group \(\mathrm{SL}(2, \mathbb{Z})\) is isomorphic to the amalgamated product \(\mathbb{Z}/4\mathbb{Z} *_{\mathbb{Z}/2\mathbb{Z}} \mathbb{Z}/6\mathbb{Z}\), where the amalgamation identifies the unique element of order 2 in each factor. By the Seifert-van Kampen theorem, if \(M = M_1 \cup M_2\) with \(M_1 \cap M_2\) connected, then \(\pi_1(M) = \pi_1(M_1) *_{\pi_1(M_1 \cap M_2)} \pi_1(M_2)\).

The structure of amalgamated products is revealed by normal forms:

Theorem 3.3 (Normal Form Theorem for Amalgams). Every element of \(A *_C B\) can be uniquely written in the form \[ c \cdot a_1 b_1 a_2 b_2 \cdots a_n b_n \] where \(c \in C\), each \(a_i\) is a representative of a non-trivial coset of \(C\) in \(A\), and each \(b_i\) is a representative of a non-trivial coset of \(C\) in \(B\) (with the convention that the sequence may begin or end with an \(a\)-term or a \(b\)-term). In particular, the natural maps \(A \to A *_C B\) and \(B \to A *_C B\) are injective.

3.3 HNN Extensions

The second fundamental construction in Bass-Serre theory is the HNN extension, named after Higman, Neumann, and Neumann who introduced it in 1949.

Let \(A\) be a group, \(C, C' \le A\) subgroups, and \(\varphi: C \xrightarrow{\sim} C'\) an isomorphism. The HNN extension of \(A\) relative to \(\varphi\) is the group \[ A *_\varphi = \langle A, t \mid t c t^{-1} = \varphi(c) \text{ for all } c \in C \rangle. \] The element \(t\) is called the stable letter, and \(A\) is the base group.

Theorem 3.4 (Britton's Lemma). Let \(G = A *_\varphi\) be an HNN extension. A word \[ a_0 t^{\varepsilon_1} a_1 t^{\varepsilon_2} \cdots t^{\varepsilon_n} a_n \] with \(a_i \in A\), \(\varepsilon_i \in \{\pm 1\}\), and \(n \ge 1\) represents the identity in \(G\) only if it contains a pinch: a subword \(t a_i t^{-1}\) with \(a_i \in C\), or \(t^{-1} a_i t\) with \(a_i \in C'\). In particular, the natural map \(A \to A *_\varphi\) is injective.

Example 3.5. The Baumslag-Solitar groups \(\mathrm{BS}(m,n) = \langle a, b \mid b a^m b^{-1} = a^n \rangle\) are HNN extensions of \(\mathbb{Z}\) where the stable letter \(b\) conjugates the subgroup \(m\mathbb{Z}\) to \(n\mathbb{Z}\). For \(m = n = 1\), this is \(\mathbb{Z}^2\). For \(|m| \neq |n|\), the group \(\mathrm{BS}(m,n)\) is a key source of examples and counterexamples in geometric group theory; for instance, \(\mathrm{BS}(1,2)\) is solvable but not virtually nilpotent, so it has exponential growth.

3.4 Graphs of Groups

Both amalgamated free products and HNN extensions are special cases of a more general construction: the fundamental group of a graph of groups.

A graph of groups \(\mathcal{G}\) consists of:

An underlying graph \(\Gamma\) (possibly with loops and multiple edges).
For each vertex \(v\) of \(\Gamma\), a vertex group \(G_v\).
For each oriented edge \(e\) of \(\Gamma\) (from \(\alpha(e)\) to \(\omega(e)\)), an edge group \(G_e\) together with injective homomorphisms \(\varphi_e^-: G_e \hookrightarrow G_{\alpha(e)}\) and \(\varphi_e^+: G_e \hookrightarrow G_{\omega(e)}\).

The fundamental group of a graph of groups \(\mathcal{G}\) (with respect to a chosen basepoint or maximal subtree \(T\) of \(\Gamma\)) is the group \(\pi_1(\mathcal{G}, T)\) generated by all vertex groups \(G_v\) and stable letters \(t_e\) (one for each edge \(e\) not in \(T\)), subject to the relations:

All relations within each vertex group \(G_v\).
\(\varphi_e^-(c) = \varphi_e^+(c)\) for each edge \(e\) in \(T\) and \(c \in G_e\).
\(t_e \varphi_e^-(c) t_e^{-1} = \varphi_e^+(c)\) for each edge \(e\) not in \(T\) and \(c \in G_e\).

The case where \(\Gamma\) is a single edge gives an amalgamated free product, and the case where \(\Gamma\) has one vertex and one loop edge gives an HNN extension.

3.5 The Bass-Serre Tree

The fundamental theorem of Bass-Serre theory establishes a tight correspondence between graphs of groups and group actions on trees.

Theorem 3.6 (Bass-Serre). Let \(\mathcal{G}\) be a graph of groups with underlying graph \(\Gamma\), and let \(G = \pi_1(\mathcal{G})\). Then there exists a tree \(T\) (the Bass-Serre tree) on which \(G\) acts without inversions such that the quotient graph \(G \backslash T\) is isomorphic to \(\Gamma\), and the vertex and edge stabilizers are conjugates of the corresponding vertex and edge groups.

Conversely, if a group \(G\) acts on a tree \(T\) without inversions, then \(G\) is the fundamental group of the graph of groups \(\mathcal{G}\) whose underlying graph is \(G \backslash T\), with vertex and edge groups being the stabilizers of lifts of vertices and edges.

This correspondence is enormously powerful: it translates questions about the algebraic structure of group splittings into questions about group actions on trees.

Example 3.7. The group \(\mathrm{SL}(2, \mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z} *_{\mathbb{Z}/2\mathbb{Z}} \mathbb{Z}/6\mathbb{Z}\) acts on its Bass-Serre tree, which is a tree where vertices of one type have valence 2 and vertices of the other type have valence 3. This is the dual tree to the Farey tessellation of the hyperbolic plane.

3.6 Stallings’ Theorem on Ends of Groups

The number of ends of a group is a fundamental quasi-isometry invariant. Informally, it measures how many “directions to infinity” the Cayley graph has.

Let \(G\) be a finitely generated group with Cayley graph \(\mathrm{Cay}(G, S)\). The number of ends \(e(G)\) is the supremum over compact subsets \(K \subseteq \mathrm{Cay}(G,S)\) of the number of unbounded connected components of \(\mathrm{Cay}(G,S) \setminus K\).

Proposition 3.8. A finitely generated group has \(0, 1, 2,\) or infinitely many ends. Specifically:

\(e(G) = 0\) if and only if \(G\) is finite.
\(e(G) = 2\) if and only if \(G\) is virtually \(\mathbb{Z}\).
\(e(G) = 1\) or \(e(G) = \infty\) in all other cases.

Stallings’ theorem characterizes groups with more than one end, establishing a profound connection between this topological invariant and the algebraic structure:

Theorem 3.9 (Stallings, 1968; 1971). A finitely generated group \(G\) has more than one end if and only if one of the following holds:

\(G\) splits as an amalgamated free product \(A *_C B\) where \(C\) is finite and \(A \neq C \neq B\).
\(G\) splits as an HNN extension \(A *_\varphi\) where the associated subgroups are finite.

In short, \(G\) has more than one end if and only if \(G\) splits over a finite subgroup.

Stallings’ original proof used topological methods. A more modern proof proceeds by constructing an action of \(G\) on a tree from the combinatorics of the Cayley graph: the ends structure gives rise to a “structure tree” on which \(G\) acts with finite edge stabilizers, and Bass-Serre theory converts this into a splitting.

Remark 3.10. Stallings' theorem can be iterated: if a group splits over a finite subgroup, we can ask whether the factors split further. This leads to the notion of accessibility. A finitely presented group \(G\) is accessible if this process terminates — that is, there is an upper bound on the number of splittings over finite subgroups. Dunwoody (1985) proved that all finitely presented groups are accessible. However, Dunwoody also showed that this fails for finitely generated groups: there exist finitely generated groups that are not accessible.

Chapter 4: Hyperbolic Groups

Hyperbolic groups, introduced by Gromov in his seminal 1987 essay Hyperbolic groups, are perhaps the most important class of groups in geometric group theory. They capture the coarse geometry of negatively curved spaces — including classical hyperbolic space, fundamental groups of negatively curved manifolds, and many combinatorially defined groups — in a purely metric framework that is robust under quasi-isometry.

4.1 Gromov Hyperbolicity

Let \((X, d)\) be a geodesic metric space. A geodesic triangle \(\Delta = \Delta(x,y,z)\) consists of three points \(x, y, z \in X\) and three geodesic segments \([x,y], [y,z], [z,x]\). The triangle is \(\delta\)-slim (or \(\delta\)-thin) if each side is contained in the \(\delta\)-neighborhood of the union of the other two sides: \[ [x,y] \subseteq \mathcal{N}_\delta([y,z] \cup [z,x]), \quad \text{and cyclically.} \] The space \(X\) is \(\delta\)-hyperbolic if every geodesic triangle in \(X\) is \(\delta\)-slim. A geodesic metric space is Gromov hyperbolic (or simply hyperbolic) if it is \(\delta\)-hyperbolic for some \(\delta \ge 0\).

There are several equivalent formulations of hyperbolicity. One useful characterization uses the Gromov product:

Let \((X, d)\) be a metric space. The Gromov product of \(x, y \in X\) with respect to a basepoint \(w \in X\) is \[ (x \cdot y)_w = \frac{1}{2}(d(x,w) + d(y,w) - d(x,y)). \]

The Gromov product measures how long geodesics from \(w\) to \(x\) and from \(w\) to \(y\) “fellow travel” before diverging. In a tree, \((x \cdot y)_w\) is exactly the distance from \(w\) to the branch point of the geodesics \([w,x]\) and \([w,y]\).

Proposition 4.1. A geodesic metric space \(X\) is Gromov hyperbolic if and only if there exists \(\delta \ge 0\) such that for all \(x, y, z, w \in X\), \[ (x \cdot y)_w \ge \min\{(x \cdot z)_w, (y \cdot z)_w\} - \delta. \] This formulation (the four-point condition) does not require \(X\) to be geodesic, so it extends the notion of hyperbolicity to arbitrary metric spaces.

A finitely generated group \(G\) is hyperbolic (or word-hyperbolic or Gromov hyperbolic) if its Cayley graph (with respect to some, equivalently any, finite generating set) is a Gromov hyperbolic metric space.

Since hyperbolicity is a quasi-isometry invariant of geodesic metric spaces, this definition is independent of the choice of generating set.

4.2 Examples of Hyperbolic Groups

Example 4.2. The following groups are hyperbolic:

Finite groups: trivially \(0\)-hyperbolic.
Free groups: The Cayley graph of \(F_n\) is a tree, which is \(0\)-hyperbolic (geodesic triangles are tripods).
Surface groups: The fundamental group \(\pi_1(\Sigma_g)\) for \(g \ge 2\) is hyperbolic, since it acts properly cocompactly on the hyperbolic plane \(\mathbb{H}^2\), which is \(\delta\)-hyperbolic.
Fundamental groups of closed negatively curved manifolds: by the Milnor-Schwarz lemma and the fact that simply connected manifolds of pinched negative curvature are Gromov hyperbolic.
Small cancellation groups satisfying \(C'(1/6)\): as proved by Greendlinger's lemma and its consequences.
Random groups: In the Gromov density model, a random group at density \(d < 1/2\) is hyperbolic with overwhelming probability (Gromov, Ollivier).

Example 4.3 (Non-examples). The following groups are not hyperbolic:

\(\mathbb{Z}^2\) (and any group containing \(\mathbb{Z}^2\) as a subgroup), since its Cayley graph contains quasi-isometric copies of \(\mathbb{R}^2\), which has geodesic triangles that are not uniformly thin.
The Baumslag-Solitar groups \(\mathrm{BS}(m,n)\) for \(|m|, |n| \ge 1\) and \((m,n) \neq (\pm 1, \pm 1)\), since they contain either \(\mathbb{Z}^2\) (when \(|m| = |n|\)) or have exponential Dehn function (when \(|m| \neq |n|\)).
Any group with an unsolvable word problem.

4.3 The Boundary at Infinity

One of the most important features of hyperbolic spaces is the existence of a well-defined boundary at infinity, which carries a rich topological and metric structure.

Let \(X\) be a proper Gromov hyperbolic space. The boundary at infinity (or Gromov boundary) \(\partial_\infty X\) is defined as the set of equivalence classes of geodesic rays, where two rays \(\gamma_1, \gamma_2: [0, \infty) \to X\) are equivalent if they have finite Hausdorff distance: \(\sup_t d(\gamma_1(t), \gamma_2(t)) < \infty\).

The boundary \(\partial_\infty X\) can be topologized so that the compactification \(\overline{X} = X \cup \partial_\infty X\) is compact and metrizable. The topology is characterized by the property that a sequence of points \(x_n \in X\) converges to \(\xi \in \partial_\infty X\) if and only if \((x_n \cdot \xi)_w \to \infty\) for some (equivalently, any) basepoint \(w\).

Example 4.4.

For a tree \(T\), the boundary \(\partial_\infty T\) is a Cantor set (if \(T\) is regular of valence \(\ge 3\)) or consists of two points (if \(T\) is a line).
For the hyperbolic plane \(\mathbb{H}^2\), the boundary \(\partial_\infty \mathbb{H}^2 \cong S^1\) is the circle at infinity.
For hyperbolic \(n\)-space \(\mathbb{H}^n\), the boundary is \(\partial_\infty \mathbb{H}^n \cong S^{n-1}\).

Proposition 4.5. The boundary at infinity is a quasi-isometry invariant: if \(f: X \to Y\) is a quasi-isometry between proper Gromov hyperbolic spaces, then \(f\) induces a homeomorphism \(\partial_\infty f: \partial_\infty X \to \partial_\infty Y\).

For a hyperbolic group \(G\), we define \(\partial_\infty G = \partial_\infty \mathrm{Cay}(G, S)\), which is well-defined up to homeomorphism. The group \(G\) acts on \(\partial_\infty G\) by homeomorphisms, and this action encodes significant algebraic information.

Theorem 4.6. Let \(G\) be a hyperbolic group. Then:

\(\partial_\infty G = \emptyset\) if and only if \(G\) is finite.
\(\partial_\infty G\) consists of exactly two points if and only if \(G\) is virtually \(\mathbb{Z}\).
If \(G\) is one-ended (and infinite, non-virtually-cyclic), then \(\partial_\infty G\) is connected and locally connected.

4.4 Dehn’s Algorithm and the Linear Isoperimetric Inequality

The algebraic and algorithmic consequences of hyperbolicity are striking. The following theorem characterizes hyperbolic groups via their Dehn functions.

Theorem 4.7. A finitely presented group \(G\) is hyperbolic if and only if it satisfies a linear isoperimetric inequality: there exists a constant \(K\) such that the Dehn function satisfies \(\delta(n) \le Kn\) for all \(n\).

We sketch the key ideas. Suppose \(G\) is \(\delta\)-hyperbolic and \(w\) is a word of length \(n\) representing the identity. Consider a geodesic triangle in the Cayley graph with vertices at \(e\), the point reached after reading the first \(\lfloor n/2 \rfloor\) letters, and some intermediate point. The \(\delta\)-slim condition implies that the loop \(w\) can be "cut" into smaller loops whose total area is controlled. Iterating this process, one can fill the loop with area at most \(Kn\) for some constant \(K\) depending on \(\delta\) and the presentation.

Conversely, if \(G\) satisfies a linear isoperimetric inequality, one shows that geodesic triangles in the Cayley graph are uniformly thin by analyzing the geometry of van Kampen diagrams with linear area. \(\blacksquare\)

The linear isoperimetric inequality has an immediate algorithmic consequence:

Theorem 4.8 (Dehn's Algorithm for Hyperbolic Groups). Let \(G\) be a hyperbolic group with presentation \(\langle X \mid R \rangle\). Then there exists a finite set of Dehn relators — words \(uv^{-1}\) where \(uv\) is a relator (or consequence of relators) and \(|u| > |v|\) — such that the following algorithm solves the word problem: given a word \(w\), repeatedly search for and replace any subword \(u\) by the corresponding shorter word \(v\). The word represents the identity if and only if it reduces to the empty word. This terminates in at most \(|w|\) steps.

Corollary 4.9. The word problem for hyperbolic groups is solvable in linear time (in the length of the input word).

Theorem 4.10. The conjugacy problem for hyperbolic groups is solvable.

The proof of the conjugacy problem is considerably more involved and uses the classification of isometries of hyperbolic spaces into elliptic and hyperbolic types, along with the structure of centralizers.

4.5 The Rips Complex and Subgroups

Let \(G\) be a group with a finite generating set \(S\), and let \(d \ge 0\). The Rips complex (or Vietoris-Rips complex) \(P_d(G, S)\) is the simplicial complex whose vertex set is \(G\) and whose simplices are finite subsets of \(G\) of diameter at most \(d\) in the word metric.

Theorem 4.11 (Rips). For a hyperbolic group \(G\), the Rips complex \(P_d(G, S)\) is contractible for all sufficiently large \(d\). Moreover, \(G\) acts properly and cocompactly on \(P_d(G, S)\) by simplicial automorphisms. In particular, every hyperbolic group admits a finite classifying space \(BG\) (equivalently, \(G\) is of type \(F_\infty\)).

This has strong consequences for the cohomological properties of hyperbolic groups. Regarding subgroups:

Theorem 4.12. Let \(G\) be a hyperbolic group. Then:

Every abelian subgroup of \(G\) is virtually cyclic.
Every solvable subgroup of \(G\) is virtually cyclic.
\(G\) contains no subgroup isomorphic to \(\mathbb{Z}^2\).
If \(G\) is torsion-free, then every non-trivial abelian subgroup is infinite cyclic.

We prove (1). Let \(A\) be an abelian subgroup of \(G\). If \(A\) is finite, it is virtually cyclic. If \(A\) contains an element \(a\) of infinite order, then \(\langle a \rangle \cong \mathbb{Z}\) and \(\langle a \rangle\) is a quasi-convex subgroup (this requires a separate argument). The key observation is that in a \(\delta\)-hyperbolic space, two bi-infinite quasi-geodesics that remain at bounded Hausdorff distance from each other must fellow travel (be at bounded distance from the same bi-infinite geodesic). If \(b \in A\) is another element of infinite order, then since \(ab = ba\), the axes of \(a\) and \(b\) in the Cayley graph must be at bounded Hausdorff distance (the axis of \(a\) is quasi-preserved by the action of \(b\), and vice versa). This forces \(b\) to be virtually a power of \(a\): specifically, some powers \(a^m\) and \(b^n\) coincide. Hence \(A\) is virtually \(\mathbb{Z}\). \(\blacksquare\)

4.6 Quasiconvex Subgroups

Let \(G\) be a hyperbolic group and \(H \le G\) a subgroup. We say \(H\) is quasiconvex (or \(C\)-quasiconvex) if there exists \(C \ge 0\) such that every geodesic in \(\mathrm{Cay}(G, S)\) between elements of \(H\) lies within the \(C\)-neighborhood of \(H\).

Quasiconvex subgroups are the “geometrically well-behaved” subgroups of hyperbolic groups. They enjoy many of the same properties as the ambient group.

Proposition 4.13. Let \(G\) be a hyperbolic group and \(H \le G\) a quasiconvex subgroup. Then:

\(H\) is finitely generated and hyperbolic.
The inclusion \(H \hookrightarrow G\) is a quasi-isometric embedding.
The word problem for \(H\) is solvable (in the generating set inherited from \(G\)).
The intersection of two quasiconvex subgroups is quasiconvex.

However, not all subgroups of hyperbolic groups are quasiconvex. The subgroup structure of hyperbolic groups can be extremely complex:

Theorem 4.14 (Rips). For every finitely presented group \(Q\), there exists a hyperbolic group \(G\) and a normal subgroup \(N \trianglelefteq G\) such that \(G/N \cong Q\). In particular, hyperbolic groups can have finitely generated normal subgroups with unsolvable word problem.

This remarkable result, proved by a sophisticated small cancellation construction, shows that while hyperbolic groups themselves have very good algorithmic properties, their subgroups can be arbitrarily complicated.

Chapter 5: CAT(0) Spaces and Groups

While hyperbolic groups capture the coarse geometry of negative curvature, CAT(0) spaces and groups provide a framework for non-positive curvature — a broader and in many ways richer theory. The notion of CAT(0) geometry, named by Gromov in honor of Cartan, Alexandrov, and Toponogov, axiomatizes the global geometry of simply connected Riemannian manifolds of non-positive sectional curvature through comparison geometry.

5.1 CAT(\(\kappa\)) Comparison Geometry

The key idea is to compare geodesic triangles in a given metric space with triangles in model spaces of constant curvature.

For \(\kappa \in \mathbb{R}\), the model space \(M^2_\kappa\) is the simply connected complete Riemannian 2-manifold of constant curvature \(\kappa\):

\(M^2_0 = \mathbb{R}^2\) (Euclidean plane), or more generally \(\mathbb{R}^n\).
\(M^2_\kappa = S^2(1/\sqrt{\kappa})\) for \(\kappa > 0\) (sphere of radius \(1/\sqrt{\kappa}\)).
\(M^2_\kappa = \mathbb{H}^2(-1/\sqrt{-\kappa})\) for \(\kappa < 0\) (hyperbolic plane of curvature \(\kappa\)).

The diameter of \(M^2_\kappa\) is \(D_\kappa = \pi / \sqrt{\kappa}\) if \(\kappa > 0\), and \(\infty\) if \(\kappa \le 0\).

Let \((X, d)\) be a geodesic metric space, \(\kappa \in \mathbb{R}\), and let \(\Delta = \Delta(x,y,z)\) be a geodesic triangle in \(X\) with perimeter less than \(2D_\kappa\). A comparison triangle is a triangle \(\overline{\Delta} = \Delta(\bar{x}, \bar{y}, \bar{z})\) in \(M^2_\kappa\) with the same side lengths: \(d_{M^2_\kappa}(\bar{x}, \bar{y}) = d(x,y)\), etc. (Such a comparison triangle exists and is unique up to isometry by the law of cosines in \(M^2_\kappa\).)

For a point \(p\) on a side \([x,y]\) of \(\Delta\), the comparison point \(\bar{p}\) is the point on \([\bar{x}, \bar{y}]\) with \(d(\bar{x}, \bar{p}) = d(x, p)\).

\[ d(p, q) \le d_{M^2_\kappa}(\bar{p}, \bar{q}). \]

In words: triangles in a CAT(\(\kappa\)) space are “thinner” than triangles of the same side lengths in the model space \(M^2_\kappa\). For \(\kappa = 0\), this means triangles are thinner than Euclidean triangles — a global non-positive curvature condition.

5.2 CAT(0) Spaces (Hadamard Spaces)

CAT(0) spaces are the most important case for geometric group theory. They are also known as Hadamard spaces when they are complete.

Theorem 5.1. Let \(X\) be a CAT(0) space. Then:

Unique geodesics: For any two points \(x, y \in X\), there is a unique geodesic segment \([x,y]\).
Convexity of the distance function: If \(\gamma, \sigma: [0,1] \to X\) are geodesics, then the function \(t \mapsto d(\gamma(t), \sigma(t))\) is convex.
Projection to convex subsets: For any non-empty closed convex subset \(C \subseteq X\) and any point \(x \in X\), there exists a unique point \(\pi_C(x) \in C\) nearest to \(x\), and the projection \(\pi_C: X \to C\) is a distance-non-increasing retraction.
Fixed point theorem (Bruhat-Tits)): If a group \(G\) acts on \(X\) by isometries and has a bounded orbit, then \(G\) has a fixed point.

We prove (2). Let \(\gamma\) be the geodesic from \(x_0\) to \(x_1\) and \(\sigma\) the geodesic from \(y_0\) to \(y_1\). Set \(f(t) = d(\gamma(t), \sigma(t))\). Consider the triangle \(\Delta(x_0, x_1, y_1)\) with comparison triangle \(\overline{\Delta}\). The CAT(0) condition gives \(d(\gamma(t), y_1) \le d(\bar{\gamma}(t), \bar{y}_1)\). Applying the CAT(0) condition again in the triangle \(\Delta(x_0, y_0, y_1)\) and using the convexity of the distance function in \(\mathbb{R}^2\) (where it holds with equality for geodesics), we obtain \(f(t) \le (1-t) f(0) + t f(1)\). \(\blacksquare\)

Example 5.2.

Euclidean space \(\mathbb{R}^n\) is CAT(0) (and CAT(\(\kappa\)) for all \(\kappa \ge 0\)).
Hyperbolic space \(\mathbb{H}^n\) is CAT(\(\kappa\)) for all \(\kappa \le 0\), hence also CAT(0).
Trees are CAT(0) (in fact, they are CAT(\(\kappa\)) for all \(\kappa\)).
Simply connected Riemannian manifolds of sectional curvature \(\le 0\) are CAT(0) (Cartan-Hadamard theorem).
Products of CAT(0) spaces are CAT(0).

A fundamental tool for constructing CAT(0) spaces from smaller pieces is:

Theorem 5.3 (Cartan-Hadamard Theorem for Metric Spaces). Let \(X\) be a complete connected metric space such that every point has a CAT(0) neighborhood. Then the universal cover \(\widetilde{X}\) (with the induced length metric) is a CAT(0) space.

This gives a local-to-global principle: to show a space is CAT(0), it suffices to verify the CAT(0) condition locally (provided the space is simply connected).

5.3 The Flat Torus Theorem

Theorem 5.4 (Flat Torus Theorem). Let \(G\) be a group acting properly and cocompactly by isometries on a CAT(0) space \(X\). If \(G\) contains a subgroup \(A \cong \mathbb{Z}^n\), then \(X\) contains an isometrically embedded copy of \(\mathbb{R}^n\) that is \(A\)-invariant, on which \(A\) acts by translations, and the quotient \(A \backslash \mathbb{R}^n\) is a flat \(n\)-torus.

For each non-trivial element \(a \in A\), the set \(\mathrm{Min}(a) = \{x \in X : d(x, ax) = \inf_y d(y, ay)\}\) is a non-empty closed convex subset of \(X\) (non-emptiness uses properness and cocompactness; convexity uses the convexity of the displacement function in CAT(0) spaces). Moreover, \(\mathrm{Min}(a)\) splits isometrically as \(C_a \times \mathbb{R}\), where the \(\mathbb{R}\)-factor is the axis of \(a\). Since \(A\) is abelian, for any \(a, b \in A\), the element \(b\) preserves \(\mathrm{Min}(a)\) (because \(bab^{-1} = a\)). Iterating this argument for a basis \(a_1, \ldots, a_n\) of \(A \cong \mathbb{Z}^n\), one finds a convex subspace isometric to \(\mathbb{R}^n\) on which \(A\) acts by translations. \(\blacksquare\)

The flat torus theorem has important consequences for understanding the algebraic structure of CAT(0) groups. For instance, it implies that every abelian subgroup of a CAT(0) group that acts properly and cocompactly is finitely generated (of rank at most the dimension of the CAT(0) space).

5.4 CAT(0) Cube Complexes

CAT(0) cube complexes have emerged as one of the most important and versatile tools in modern geometric group theory, playing a central role in Agol’s proof of the virtual Haken conjecture and Wise’s work on residual finiteness.

A cube complex is a cell complex in which each cell is isometric to a Euclidean cube \([0,1]^n\) (for varying \(n\)) and the attaching maps are isometries onto faces. A cube complex is CAT(0) if it is simply connected and satisfies the Gromov link condition: the link of every vertex is a flag complex (i.e., any collection of pairwise-adjacent vertices spans a simplex).

Theorem 5.5 (Gromov). A simply connected cube complex \(X\) is CAT(0) (with the intrinsic metric obtained by declaring each cube to be Euclidean) if and only if the link of every vertex is a flag complex.

This combinatorial criterion for the CAT(0) condition is remarkably useful in practice, as it reduces a metric condition to a purely combinatorial one.

5.5 Hyperplanes and Sageev’s Construction

The geometry of CAT(0) cube complexes is governed by their hyperplanes, a structure with no direct analogue in general CAT(0) spaces.

Let \(X\) be a CAT(0) cube complex. A midcube of an \(n\)-cube \([0,1]^n\) is a subset obtained by restricting one coordinate to \(1/2\). A hyperplane in \(X\) is a connected subspace \(H\) such that for each cube \(C\) of \(X\), \(H \cap C\) is either empty or a midcube.

Proposition 5.6. In a CAT(0) cube complex \(X\), each hyperplane \(H\) is itself a CAT(0) cube complex (of dimension one less). Moreover:

\(H\) separates \(X\) into exactly two connected components (halfspaces).
\(H\) is convex in \(X\).
Two hyperplanes either are disjoint, cross (intersect transversally), or osculate (share a cube but do not cross).
The combinatorial distance between two vertices equals the number of hyperplanes separating them.

Sageev’s construction (1995) provides a powerful machine for building CAT(0) cube complexes from group actions:

Theorem 5.7 (Sageev, 1995). Let \(G\) be a finitely generated group and let \(H \le G\) be a codimension-1 subgroup (i.e., \(H\) separates the Cayley graph into at least two deep components). Then \(G\) acts on a CAT(0) cube complex \(X\) such that \(H\) stabilizes a hyperplane.

This construction has been spectacularly successful. Wise and Agol used it to prove:

Theorem 5.8 (Agol, 2012; building on Wise). Every hyperbolic group that acts properly and cocompactly on a CAT(0) cube complex is virtually special (in the sense of Haglund-Wise), and in particular is linear over \(\mathbb{Z}\), residually finite, and virtually torsion-free.

5.6 CAT(0) Groups and Their Properties

A group \(G\) is a CAT(0) group if it acts properly and cocompactly by isometries on a CAT(0) space.

Theorem 5.9. Let \(G\) be a CAT(0) group. Then:

\(G\) is finitely presented.
\(G\) has solvable word problem.
\(G\) has a finite-dimensional classifying space \(BG\) (in particular, \(G\) is of type \(FP_\infty\)).
Every abelian subgroup of \(G\) is finitely generated.
\(G\) satisfies a quadratic isoperimetric inequality (Dehn function \(\delta(n) \preceq n^2\)).
\(G\) has finitely many conjugacy classes of finite subgroups.

Statement (1) follows from the Milnor-Schwarz lemma (finite generation) and the fact that \(G\) acts on a contractible space (the CAT(0) space) with compact quotient (giving a finite presentation via the presentation complex of the quotient). Statement (2) follows from (5): a quadratic Dehn function is computable, giving a solution to the word problem. Statement (3) follows from the Rips-complex-type construction: the quotient of the CAT(0) space by the action is a compact orbispace serving as a model for \(BG\). Statement (4) follows from the flat torus theorem. Statement (5) is a deep result using the geometry of geodesic fillings in CAT(0) spaces. \(\blacksquare\)

5.7 The Visual Boundary of CAT(0) Spaces

Let \(X\) be a complete CAT(0) space. The visual boundary (or boundary at infinity) \(\partial_\infty X\) is the set of equivalence classes of geodesic rays, where two rays are equivalent if they have bounded Hausdorff distance (equivalently, if they are asymptotic: \(\sup_t d(\gamma_1(t), \gamma_2(t)) < \infty\)).

Unlike in the hyperbolic case, the visual boundary of a CAT(0) space is not a quasi-isometry invariant — there exist quasi-isometric CAT(0) spaces with non-homeomorphic boundaries. This is one of the key differences between CAT(0) and hyperbolic geometry.

Example 5.10. The boundary of \(\mathbb{R}^n\) is the sphere \(S^{n-1}\). The boundary of a regular tree of valence \(\ge 3\) is a Cantor set. The boundary of \(\mathbb{H}^n\) is \(S^{n-1}\), the same as \(\mathbb{R}^n\) — but the two spaces are not quasi-isometric, showing that the boundary alone does not determine the quasi-isometry type in the CAT(0) setting.

5.8 Right-Angled Artin Groups and Coxeter Groups

Two important families of CAT(0) groups arise from combinatorial data encoded in graphs.

Let \(\Gamma\) be a finite simplicial graph with vertex set \(V\) and edge set \(E\). The right-angled Artin group (RAAG) associated to \(\Gamma\) is \[ A_\Gamma = \langle v \in V \mid [v, w] = 1 \text{ for } \{v, w\} \in E \rangle. \] That is, generators commute if and only if they are connected by an edge in \(\Gamma\).

RAAGs interpolate between free groups (when \(\Gamma\) has no edges) and free abelian groups (when \(\Gamma\) is complete). They act properly and cocompactly on CAT(0) cube complexes (the Salvetti complex), so they are CAT(0) groups.

Let \((W, S)\) be a Coxeter system: \(W\) is a group generated by a finite set \(S\) of involutions with presentation \[ W = \langle S \mid (st)^{m(s,t)} = 1 \text{ for } s, t \in S \text{ with } m(s,t) < \infty \rangle \] where \(m(s,s) = 1\) and \(m(s,t) = m(t,s) \ge 2\) for \(s \neq t\).

Coxeter groups act on the Davis complex, a CAT(0) cube complex (or more generally a CAT(0) piecewise Euclidean cell complex), making them CAT(0) groups. Finite Coxeter groups are precisely the finite reflection groups, and the infinite ones include fundamental groups of important geometric objects like right-angled polyhedra in hyperbolic space.

Chapter 6: Amenability and Property (T)

The notions of amenability and Kazhdan’s property (T) stand at opposite ends of a spectrum. Amenable groups — those admitting an invariant mean or satisfying the Følner condition — are characterized by having “little resistance” to averaging; they include all abelian, solvable, and finitely generated groups of subexponential growth. At the other extreme, groups with property (T) have the strongest possible rigidity: every affine isometric action on a Hilbert space has a fixed point. The interplay between these two notions has produced some of the deepest results in group theory and its applications.

6.1 Amenable Groups

The theory of amenable groups was initiated by John von Neumann in 1929, motivated by the Banach-Tarski paradox. Von Neumann identified amenability as the key property separating the groups for which paradoxical decompositions are possible from those for which they are not.

A group \(G\) is amenable if it admits a finitely additive, left-invariant probability measure on all subsets of \(G\). Equivalently, there exists a left-invariant mean: a positive linear functional \(\mu: \ell^\infty(G) \to \mathbb{R}\) with \(\mu(\mathbf{1}) = 1\) and \(\mu(g \cdot f) = \mu(f)\) for all \(g \in G\) and \(f \in \ell^\infty(G)\), where \((g \cdot f)(x) = f(g^{-1}x)\).

The existence of an invariant mean is a very strong condition, but it is most useful when combined with more geometric characterizations:

Theorem 6.1 (Følner, 1955). A finitely generated group \(G\) is amenable if and only if it satisfies the Følner condition: for every \(\varepsilon > 0\) and every finite subset \(F \subseteq G\), there exists a non-empty finite subset \(A \subseteq G\) (a Følner set) such that \[ \frac{|FA \,\triangle\, A|}{|A|} < \varepsilon. \] Equivalently, for any finite generating set \(S\), \[ \lim_{n \to \infty} \frac{|\partial A_n|}{|A_n|} = 0 \] for some sequence of finite sets \(A_n\), where \(\partial A_n = \{g \in A_n : gs \notin A_n \text{ for some } s \in S\}\) is the inner boundary.

The Følner condition says that amenable groups admit “almost-invariant” finite subsets — sets whose boundary is small relative to their volume. This is an isoperimetric condition, dual to the notion of expansion.

Example 6.2. The following groups are amenable:

Finite groups (take \(A = G\)).
\(\mathbb{Z}^n\) (take \(A = [-N, N]^n\), then \(|\partial A|/|A| \to 0\)).
All abelian groups.
All solvable groups.
All finitely generated groups of subexponential growth.
Extensions and direct limits of amenable groups.

The class of amenable groups is closed under taking subgroups, quotients, extensions, and directed unions. The smallest class of groups containing all finite and abelian groups and closed under these operations is the class of elementary amenable groups.

6.2 The Banach-Tarski Paradox and Non-Amenable Groups

Theorem 6.3 (Banach-Tarski, 1924). The unit ball in \(\mathbb{R}^3\) can be decomposed into finitely many pieces (using the Axiom of Choice) and reassembled by rigid motions into two copies of the unit ball.

The Banach-Tarski paradox is possible because the rotation group \(\mathrm{SO}(3)\) contains a free subgroup of rank 2, and free groups are not amenable. More precisely:

Theorem 6.4 (von Neumann, 1929). A group \(G\) admits a paradoxical decomposition (relative to its left-translation action on itself) if and only if \(G\) is non-amenable.

Proposition 6.5. The free group \(F_2\) is not amenable.

Let \(F_2 = \langle a, b \rangle\) and suppose \(\mu\) is a left-invariant mean on \(\ell^\infty(F_2)\). For \(x \in \{a, b\}\), let \(W(x)\) denote the set of reduced words starting with \(x\), and similarly for \(W(x^{-1})\). Then \(F_2 = \{e\} \sqcup W(a) \sqcup W(a^{-1}) \sqcup W(b) \sqcup W(b^{-1})\). We also have \(F_2 = W(a) \sqcup a W(a^{-1})\) (every element either starts with \(a\) or, after prepending \(a\), starts with a letter other than \(a\)). By left-invariance, \(\mu(W(a^{-1})) = \mu(a W(a^{-1}))\), so \(1 = \mu(W(a)) + \mu(a W(a^{-1})) = \mu(W(a)) + \mu(W(a^{-1}))\). Similarly, \(\mu(W(b)) + \mu(W(b^{-1})) = 1\). But then \(\mu(W(a)) + \mu(W(a^{-1})) + \mu(W(b)) + \mu(W(b^{-1})) = 2\), while these four sets (together with \(\{e\}\)) partition \(F_2\), so their measures should sum to at most 1. Contradiction. \(\blacksquare\)

6.3 Von Neumann’s Conjecture and Its Resolution

Von Neumann conjectured (or at least his work suggested) that a group is non-amenable if and only if it contains a non-abelian free subgroup. This became known as the von Neumann conjecture (or the von Neumann-Day problem, since Day explicitly formulated it).

Theorem 6.6 (Olshanskii, 1980). The von Neumann conjecture is false: there exist non-amenable groups that contain no non-abelian free subgroup.

Olshanskii constructed such groups using his theory of graded small cancellation (Tarski monsters — infinite groups in which every proper non-trivial subgroup is cyclic of fixed prime order). A more geometric counterexample was later given by Olshanskii and Sapir.

However, a variant of the von Neumann conjecture holds in a measure-theoretic sense:

Theorem 6.7 (Whyte, 2001). A finitely generated group is non-amenable if and only if it admits a translation-like action of a non-abelian free group by bijections.

6.4 Kazhdan’s Property (T)

Property (T) was introduced by Kazhdan in 1967 as a tool for showing that certain lattices in Lie groups are finitely generated. It has since found deep applications in ergodic theory, operator algebras, combinatorics (construction of expander graphs), and theoretical computer science.

Let \(G\) be a locally compact group. A unitary representation \((\pi, \mathcal{H})\) of \(G\) on a Hilbert space \(\mathcal{H}\) almost has invariant vectors if for every compact subset \(K \subseteq G\) and every \(\varepsilon > 0\), there exists a unit vector \(v \in \mathcal{H}\) such that \(\|\pi(g)v - v\| < \varepsilon\) for all \(g \in K\).

The group \(G\) has Kazhdan’s property (T) if every unitary representation of \(G\) that almost has invariant vectors actually has a non-zero invariant vector.

For finitely generated groups, property (T) can be characterized in terms of a generating set:

Proposition 6.8. A finitely generated group \(G\) with finite generating set \(S\) has property (T) if and only if there exists \(\varepsilon > 0\) (called a Kazhdan constant) such that for every unitary representation \((\pi, \mathcal{H})\) without non-zero invariant vectors and every unit vector \(v \in \mathcal{H}\), \[ \max_{s \in S} \|\pi(s)v - v\| \ge \varepsilon. \]

Theorem 6.9 (Fixed Point Characterization). A locally compact group \(G\) has property (T) if and only if every continuous affine isometric action of \(G\) on a Hilbert space has a fixed point.

This characterization, due to Delorme and Guichardet, makes clear the rigidity aspect of property (T): the group cannot act on a Hilbert space without a fixed point. It stands in stark contrast to amenability, which (by the Følner condition) means the group has “almost fixed points” in a certain sense.

6.5 Examples and Properties

Example 6.10.

Compact groups have property (T) (by the Peter-Weyl theorem, every unitary representation decomposes into finite-dimensional irreducible representations, and almost-invariant vectors project to invariant vectors).
\(\mathrm{SL}(n, \mathbb{Z})\) for \(n \ge 3\) has property (T). This was Kazhdan's motivating example. The proof uses the embedding of \(\mathrm{SL}(n, \mathbb{Z})\) as a lattice in \(\mathrm{SL}(n, \mathbb{R})\) and the fact that the latter has property (T) for \(n \ge 3\).
\(\mathrm{Sp}(2n, \mathbb{Z})\) for \(n \ge 2\) has property (T).
Lattices in simple Lie groups of rank \(\ge 2\) have property (T).

Example 6.11 (Non-examples).

\(\mathbb{Z}\) does not have property (T): the representation \(\pi(n) = e^{in\theta}\) on \(\mathbb{C}\) almost has invariant vectors (for small \(\theta\)) but has no non-zero invariant vector (for \(\theta \neq 0\)).
Free groups \(F_n\) (\(n \ge 1\)) do not have property (T).
Amenable infinite groups do not have property (T) (unless they are compact).
Groups that split as amalgamated free products or HNN extensions over amenable subgroups do not have property (T) (by a result of Watatani).

Proposition 6.12. Property (T) has the following permanence properties:

If \(G\) has property (T) and \(N \trianglelefteq G\) is a closed normal subgroup, then \(G/N\) has property (T).
If \(G\) has property (T) and \(H \le G\) is a lattice (discrete subgroup of finite covolume), then \(H\) has property (T).
Property (T) is inherited by quotients but not by subgroups in general.
If \(G\) is finitely generated and has property (T), then \(G\) is finitely generated (in fact, compactly generated in the locally compact case) and the abelianization \(G/[G,G]\) is finite.

6.6 Expander Graphs from Property (T)

One of the most striking applications of property (T) is the construction of expander graphs — highly connected sparse graphs with applications throughout computer science and mathematics.

A family of finite graphs \(\{G_n\}\) with \(|V(G_n)| \to \infty\) and uniformly bounded vertex degree is a family of expanders if there exists \(\varepsilon > 0\) such that for every \(n\) and every subset \(A \subseteq V(G_n)\) with \(|A| \le |V(G_n)|/2\), \[ |\partial A| \ge \varepsilon |A|, \] where \(\partial A\) is the set of edges between \(A\) and its complement.

Theorem 6.13 (Margulis, 1973). Let \(G\) be a finitely generated group with property (T), let \(S\) be a finite generating set, and let \(\{N_i\}\) be a family of finite-index normal subgroups with \([G : N_i] \to \infty\). Then the Cayley graphs \(\{\mathrm{Cay}(G/N_i, \overline{S})\}\) form a family of expanders.

The key point is that property (T) provides a uniform spectral gap. For each quotient \(G/N_i\), the Kazhdan constant \(\varepsilon\) for the pair \((G, S)\) gives a lower bound on the spectral gap of the Laplacian of \(\mathrm{Cay}(G/N_i, \overline{S})\): for any function \(f: G/N_i \to \mathbb{R}\) with mean zero, \[ \frac{1}{|S|} \sum_{s \in S} \|f - f \circ s\|_2^2 \ge \varepsilon^2 \|f\|_2^2. \] By the Cheeger inequality, this spectral gap implies the expansion property. \(\blacksquare\)

Example 6.14. Taking \(G = \mathrm{SL}(3, \mathbb{Z})\) with its standard generators and \(N_p = \ker(\mathrm{SL}(3, \mathbb{Z}) \to \mathrm{SL}(3, \mathbb{Z}/p\mathbb{Z}))\) for primes \(p\), we obtain an explicit family of expander graphs. These were the first explicit constructions of expanders, given by Margulis in 1973.

6.7 The Tension Between Amenability and Property (T)

Theorem 6.15. A locally compact group is both amenable and has property (T) if and only if it is compact.

If \(G\) is compact, then \(G\) is amenable (Haar measure provides an invariant mean) and has property (T) (by the Peter-Weyl theorem). Conversely, suppose \(G\) is amenable and has property (T). Since \(G\) has property (T), the abelianization \(G/[G,G]\) is compact. We claim \(G\) itself is compact. By amenability, the left-regular representation \(\lambda: G \to \mathcal{U}(L^2(G))\) almost has invariant vectors (this is essentially the Følner condition). By property (T), \(\lambda\) has a non-zero invariant vector \(f \in L^2(G)\). But a left-invariant \(L^2\) function on \(G\) is constant, and a non-zero constant function is in \(L^2(G)\) only if \(G\) has finite Haar measure, i.e., \(G\) is compact. \(\blacksquare\)

This theorem beautifully illustrates the diametrically opposed natures of amenability and property (T): amenability is a “softness” condition (almost-invariant vectors exist), while property (T) is a “rigidity” condition (almost-invariant vectors must be close to truly invariant ones). The only groups satisfying both are compact, which are simultaneously “soft” (finite measure) and “rigid” (complete reducibility of representations).

Chapter 7: Boundaries, Rigidity, and Modern Directions

The final chapter surveys some of the most powerful results and active research directions in geometric group theory. We begin with the Gromov boundary of hyperbolic groups and the Poisson boundary, proceed to the celebrated Mostow rigidity theorem and its generalizations, and conclude with an overview of modern developments including relatively hyperbolic groups, acylindrical hyperbolicity, mapping class groups, and \(\mathrm{Out}(F_n)\).

7.1 The Gromov Boundary of Hyperbolic Groups

We have already defined the boundary at infinity of a hyperbolic group in Chapter 4. Here we explore its richer structure. The Gromov boundary \(\partial_\infty G\) admits a family of visual metrics:

Let \(G\) be a hyperbolic group. A metric \(\rho\) on \(\partial_\infty G\) is a visual metric (with parameter \(a > 1\)) if there exist constants \(C_1, C_2 > 0\) such that for all \(\xi, \eta \in \partial_\infty G\), \[ C_1 a^{-(\xi \cdot \eta)_w} \le \rho(\xi, \eta) \le C_2 a^{-(\xi \cdot \eta)_w}, \] where \((\xi \cdot \eta)_w\) is the Gromov product extended to the boundary.

Visual metrics exist for all sufficiently small parameters \(a > 1\) (specifically, for \(a\) close to 1 depending on \(\delta\)). Different visual metrics (and different parameters) give quasisymmetrically equivalent metrics on \(\partial_\infty G\).

Theorem 7.1 (Paulin; Bowditch). The homeomorphism type of \(\partial_\infty G\) together with the quasisymmetry class of its visual metrics determines the quasi-isometry class of the hyperbolic group \(G\) (among hyperbolic groups).

Theorem 7.2 (Bestvina-Mess, 1991). Let \(G\) be a torsion-free hyperbolic group. Then \[ H^n(G; \mathbb{Z}G) \cong \widetilde{H}^{n-1}(\partial_\infty G; \mathbb{Z}) \] for all \(n\), where the left side is group cohomology with coefficients in the group ring. In particular, if \(G\) is a Poincaré duality group of dimension \(n\), then \(\partial_\infty G\) is a generalized \((n-1)\)-sphere (a \(\check{\text{C}}\)ech cohomology \((n-1)\)-sphere).

The Bestvina-Mess formula has important consequences for the topology of hyperbolic groups. Combined with work of Bestvina and others, it leads to:

Corollary 7.3. If \(G\) is a one-ended torsion-free hyperbolic group, then \(\partial_\infty G\) is connected and locally connected. If \(G\) is the fundamental group of a closed hyperbolic \(n\)-manifold, then \(\partial_\infty G \cong S^{n-1}\).

The Cannon conjecture asserts the converse: if \(G\) is a hyperbolic group with \(\partial_\infty G \cong S^2\), then \(G\) acts properly cocompactly on \(\mathbb{H}^3\) (equivalently, \(G\) is virtually a closed hyperbolic 3-manifold group). This remains one of the most important open problems in geometric group theory.

7.2 The Poisson Boundary

The Poisson boundary provides a measure-theoretic analogue of the topological boundary, with deep connections to harmonic analysis, random walks, and rigidity.

Let \(G\) be a countable group and \(\mu\) a probability measure on \(G\) whose support generates \(G\). The Poisson boundary \((B, \nu)\) of \((G, \mu)\) is a measure space (with \(\nu\) a \(G\)-quasi-invariant measure, known as the harmonic measure) such that:

The space of bounded \(\mu\)-harmonic functions on \(G\) is isometrically isomorphic to \(L^\infty(B, \nu)\).
For \(\mu\)-almost every sample path \((w_n)_{n \ge 0}\) of the random walk on \(G\) (where \(w_n = g_1 g_2 \cdots g_n\) with \(g_i\) independent samples from \(\mu\)), the sequence \(w_n\) converges to a point in \(B\).

Theorem 7.4 (Kaimanovich, 2000). For a hyperbolic group \(G\) with a measure \(\mu\) of finite first moment, the Poisson boundary is naturally identified with the Gromov boundary \(\partial_\infty G\) equipped with the hitting measure of the random walk.

Theorem 7.5 (Kaimanovich-Vershik, 1983). A countable group \(G\) is amenable if and only if the Poisson boundary of \((G, \mu)\) is trivial (consists of a single point) for every non-degenerate measure \(\mu\).

7.3 Mostow Rigidity

The Mostow rigidity theorem is one of the most profound results in geometry and group theory. It states that the geometry of a closed hyperbolic manifold of dimension \(\ge 3\) is completely determined by its fundamental group — a striking rigidity phenomenon that has no analogue in dimension 2 (where the Teichmüller space of a surface parameterizes a continuous family of hyperbolic structures).

Theorem 7.6 (Mostow Rigidity, 1968). Let \(M_1\) and \(M_2\) be closed hyperbolic manifolds of dimension \(n \ge 3\). If \(\pi_1(M_1) \cong \pi_1(M_2)\), then \(M_1\) is isometric to \(M_2\). Equivalently, if \(\Gamma_1, \Gamma_2 \le \mathrm{Isom}(\mathbb{H}^n)\) are cocompact lattices and \(\varphi: \Gamma_1 \to \Gamma_2\) is a group isomorphism, then \(\varphi\) is realized by conjugation by an isometry of \(\mathbb{H}^n\).

The proof proceeds through several remarkable steps:

Sketch of proof. An isomorphism \(\varphi: \Gamma_1 \to \Gamma_2\) induces a homotopy equivalence \(f: M_1 \to M_2\), which lifts to a \(\varphi\)-equivariant quasi-isometry \(\tilde{f}: \mathbb{H}^n \to \mathbb{H}^n\). Since \(\mathbb{H}^n\) is Gromov hyperbolic, \(\tilde{f}\) induces a homeomorphism \(\partial_\infty \tilde{f}: S^{n-1} \to S^{n-1}\) of the boundary sphere.

The key analytical step is showing that \(\partial_\infty \tilde{f}\) is quasiconformal. In dimension \(n \ge 3\), one then applies the following rigidity result: a quasiconformal homeomorphism of \(S^{n-1}\) that conjugates a cocompact lattice action to itself must be conformal (this is where the dimension hypothesis \(n \ge 3\) is essential — it fails in dimension 2). A conformal map of \(S^{n-1}\) extends to an isometry of \(\mathbb{H}^n\), giving the desired rigidity. \(\blacksquare\)

Remark 7.7. Mostow rigidity was extended by Prasad (1973) to the finite-volume (non-compact) case, and by Margulis to general higher-rank lattices (Margulis superrigidity). These results show that lattices in semisimple Lie groups of higher rank are extremely rigid — their algebraic structure is completely determined by their ambient Lie group, and every homomorphism to another semisimple group virtually extends to a Lie group homomorphism.

7.4 Quasi-Isometric Rigidity

Mostow rigidity can be viewed as a special case of a more general phenomenon: quasi-isometric rigidity, which asks when the quasi-isometry class of a group determines the group up to virtual isomorphism.

Two groups \(G_1, G_2\) are virtually isomorphic (or commensurable up to finite kernels) if there exist finite-index subgroups \(H_i \le G_i\) and finite normal subgroups \(N_i \trianglelefteq H_i\) such that \(H_1/N_1 \cong H_2/N_2\).

Theorem 7.8 (Sullivan, Tukia, Casson-Jungreis, Gabai). Any finitely generated group quasi-isometric to \(\mathbb{H}^n\) (\(n \ge 2\)) acts properly and cocompactly on \(\mathbb{H}^n\) by isometries. In particular, it is virtually a cocompact lattice in \(\mathrm{Isom}(\mathbb{H}^n)\).

For \(n = 2\), this is the Casson-Jungreis and Gabai theorem (proved independently around 1992): a group quasi-isometric to \(\mathbb{H}^2\) is virtually a surface group. For \(n \ge 3\), it follows from the work of Sullivan and Tukia on quasiconformal groups.

Theorem 7.9 (Stallings; Dunwoody; Papasoglu). Being virtually free is a quasi-isometry invariant: any finitely generated group quasi-isometric to a finitely generated free group is itself virtually free.

Theorem 7.10 (Eskin-Fisher-Whyte, 2012). Any finitely generated group quasi-isometric to the integral Heisenberg group \(H_3(\mathbb{Z})\) is virtually isomorphic to \(H_3(\mathbb{Z})\).

These results illustrate a general principle: for many “geometrically rigid” groups, the quasi-isometry class essentially determines the group up to virtual isomorphism.

7.5 Relatively Hyperbolic Groups

Relatively hyperbolic groups generalize hyperbolic groups by allowing for a controlled set of “non-hyperbolic” subgroups (called peripheral subgroups), analogous to how a finite-volume hyperbolic manifold is hyperbolic away from its cusps.

A finitely generated group \(G\) is hyperbolic relative to a collection of subgroups \(\{H_1, \ldots, H_k\}\) (called peripheral subgroups) if the coned-off Cayley graph — obtained from the Cayley graph of \(G\) by adding an edge of length 1 between any two elements in the same left coset of some \(H_i\) — is Gromov hyperbolic, and the pair \((G, \{H_i\})\) satisfies a bounded coset penetration condition.

There are several equivalent formulations due to Gromov, Farb, Bowditch, Osin, and Drutu-Sapir. The most geometric is Bowditch’s:

Theorem 7.11 (Bowditch). \(G\) is hyperbolic relative to \(\{H_1, \ldots, H_k\}\) if and only if \(G\) acts properly discontinuously by isometries on a proper \(\delta\)-hyperbolic space \(X\) such that:

The action is cofinite on the set of points of \(\partial_\infty X\) that are not parabolic fixed points.
The parabolic fixed points of the action are precisely the limit points of the conjugates of the \(H_i\), and each such point has stabilizer conjugate to some \(H_i\).

Example 7.12.

Fundamental groups of finite-volume hyperbolic manifolds with cusps are hyperbolic relative to their cusp subgroups (which are virtually nilpotent).
Free products \(G = A * B\) are hyperbolic relative to \(\{A, B\}\).
Limit groups (finitely generated fully residually free groups) are hyperbolic relative to their maximal non-cyclic abelian subgroups.

7.6 Acylindrically Hyperbolic Groups

Acylindrical hyperbolicity is a much broader generalization of hyperbolicity that has become central to modern geometric group theory. It was systematically developed by Osin (2016), building on earlier work of Bowditch, Bestvina-Fujiwara, and Dahmani-Guirardel-Osin.

An action of a group \(G\) on a metric space \(X\) by isometries is acylindrical if for every \(\varepsilon > 0\), there exist \(R, N > 0\) such that for all \(x, y \in X\) with \(d(x, y) \ge R\), \[ |\{g \in G : d(x, gx) \le \varepsilon \text{ and } d(y, gy) \le \varepsilon\}| \le N. \] A group \(G\) is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a Gromov hyperbolic space (where non-elementary means the limit set of any orbit in the Gromov boundary has more than two points).

Theorem 7.13 (Osin, 2016). The following are equivalent for a group \(G\) that is not virtually cyclic:

\(G\) is acylindrically hyperbolic.
\(G\) admits a non-elementary acylindrical action on a hyperbolic space.
\(G\) contains a proper infinite hyperbolically embedded subgroup.
\(G\) is not virtually cyclic and admits an action on a hyperbolic space with a WPD (weak proper discontinuity) element.

Example 7.14. The class of acylindrically hyperbolic groups is vast:

All non-elementary hyperbolic groups.
All non-elementary relatively hyperbolic groups.
Mapping class groups \(\mathrm{Mod}(\Sigma_g)\) for \(g \ge 1\) (except for a few small-complexity cases), acting on the curve complex.
\(\mathrm{Out}(F_n)\) for \(n \ge 2\), acting on the free factor complex.
Non-virtually-cyclic groups acting properly on proper CAT(0) spaces and containing a rank-one isometry.
Most 3-manifold groups (via the Bestvina-Bromberg-Fujiwara machinery).
Right-angled Artin groups whose defining graph is connected and not a complete graph.

Acylindrically hyperbolic groups share many of the strong properties of hyperbolic groups:

Theorem 7.15. Let \(G\) be an acylindrically hyperbolic group. Then:

\(G\) contains a non-degenerate hyperbolically embedded subgroup, and hence a non-abelian free subgroup (so \(G\) is non-amenable).
\(G\) is SQ-universal: every countable group embeds in a quotient of \(G\).
The bounded cohomology \(H^2_b(G; \mathbb{R})\) is infinite-dimensional.
\(G\) has no non-trivial finite normal subgroups if it has trivial finite radical.

7.7 Mapping Class Groups

Mapping class groups are a central topic at the intersection of geometric group theory, low-dimensional topology, and algebraic geometry.

Let \(\Sigma = \Sigma_{g,n}\) be an oriented surface of genus \(g\) with \(n\) punctures. The mapping class group \(\mathrm{Mod}(\Sigma)\) is the group of isotopy classes of orientation-preserving homeomorphisms of \(\Sigma\): \[ \mathrm{Mod}(\Sigma) = \pi_0(\mathrm{Homeo}^+(\Sigma)). \]

The study of mapping class groups draws deep analogies with both lattices in Lie groups and with the group \(\mathrm{Out}(F_n)\). The key geometric tool is the curve complex:

The curve complex \(\mathcal{C}(\Sigma)\) is the simplicial complex whose vertices are isotopy classes of essential simple closed curves on \(\Sigma\), with a simplex for each collection of pairwise disjoint curves.

Theorem 7.16 (Masur-Minsky, 1999). The curve complex \(\mathcal{C}(\Sigma)\) is Gromov hyperbolic.

This landmark theorem opened the door to applying the machinery of hyperbolic geometry and boundaries to the study of mapping class groups. The mapping class group acts on \(\mathcal{C}(\Sigma)\) by simplicial automorphisms, and this action is acylindrical (Bowditch, 2008).

Theorem 7.17 (Nielsen-Thurston Classification). Every mapping class \(\phi \in \mathrm{Mod}(\Sigma)\) is one of:

Periodic: \(\phi^n = \mathrm{id}\) for some \(n\).
Reducible: \(\phi\) preserves (up to isotopy) a non-empty collection of pairwise disjoint essential simple closed curves.
Pseudo-Anosov: \(\phi\) preserves a pair of transverse measured foliations, stretching one and compressing the other by a factor \(\lambda > 1\) (the stretch factor or dilatation).

Pseudo-Anosov elements are the generic and most interesting type. They act as hyperbolic (loxodromic) isometries of the curve complex and of the Teichmüller space (with the Teichmüller metric).

Theorem 7.18 (Farb-Lubotzky-Minsky, 2001). The mapping class group \(\mathrm{Mod}(\Sigma_g)\) does not have property (T) for \(g \ge 1\). However, it is not amenable for \(g \ge 2\) (it contains free subgroups). The mapping class group satisfies a quadratic isoperimetric inequality (Mosher, 1995).

7.8 \(\mathrm{Out}(F_n)\) and Culler-Vogtmann Outer Space

The group of outer automorphisms of the free group, \(\mathrm{Out}(F_n) = \mathrm{Aut}(F_n) / \mathrm{Inn}(F_n)\), is the algebraic analogue of the mapping class group, with the role of the surface played by a graph.

Outer space \(\mathrm{CV}_n\), introduced by Culler and Vogtmann (1986), is the space of marked metric graph structures on the rose \(R_n\) (a graph with one vertex and \(n\) loop edges). More precisely, points of \(\mathrm{CV}_n\) are equivalence classes of pairs \((\Gamma, \alpha)\) where \(\Gamma\) is a finite metric graph with fundamental group \(F_n\) and \(\alpha: R_n \to \Gamma\) is a homotopy equivalence (the marking), with total volume normalized to 1.

Outer space is a contractible space on which \(\mathrm{Out}(F_n)\) acts properly (but not cocompactly). It plays a role for \(\mathrm{Out}(F_n)\) analogous to the role of Teichmüller space for mapping class groups, and of symmetric spaces for arithmetic groups.

Theorem 7.19 (Culler-Vogtmann, 1986). Outer space \(\mathrm{CV}_n\) is contractible. Its quotient \(\mathrm{Out}(F_n) \backslash \mathrm{CV}_n\) is compact (but has orbifold singularities). In particular, \(\mathrm{Out}(F_n)\) is of type VFL (virtual finite length).

The analogy between \(\mathrm{Out}(F_n)\) and mapping class groups extends to the classification of elements (analogous to the Nielsen-Thurston classification) and to the construction of hyperbolic complexes:

Theorem 7.20 (Bestvina-Feighn, 2014). The free factor complex of \(F_n\) is Gromov hyperbolic.

Theorem 7.21 (Bestvina-Feighn, 2000). \(\mathrm{Out}(F_n)\) satisfies an exponential isoperimetric inequality for \(n \ge 3\). For \(n = 2\), \(\mathrm{Out}(F_2) \cong \mathrm{GL}(2, \mathbb{Z})\) is virtually free.

7.9 Connections to Low-Dimensional Topology

Geometric group theory has deep connections to 3-manifold topology, catalyzed by Thurston’s geometrization program and its completion by Perelman (2003).

Theorem 7.22 (Thurston Geometrization, proved by Perelman). Every closed orientable 3-manifold can be decomposed along spheres and tori into pieces, each of which admits one of eight geometric structures (the Thurston geometries): \(\mathbb{S}^3\), \(\mathbb{R}^3\), \(\mathbb{H}^3\), \(\mathbb{S}^2 \times \mathbb{R}\), \(\mathbb{H}^2 \times \mathbb{R}\), \(\widetilde{\mathrm{SL}(2,\mathbb{R})}\), Nil, Sol.

From the viewpoint of geometric group theory, the most interesting case is the hyperbolic one: closed hyperbolic 3-manifolds. Their fundamental groups are hyperbolic groups, and the interplay between group-theoretic and 3-manifold properties has driven much of the field.

Theorem 7.23 (Agol, 2012; building on Wise, Kahn-Markovic, and many others). Every closed hyperbolic 3-manifold is virtually Haken: it has a finite cover containing an embedded incompressible surface. Moreover, its fundamental group is virtually special (in the sense of Haglund-Wise), and hence is linear over \(\mathbb{Z}\), residually finite, and LERF (locally extended residually finite).

Agol’s theorem resolved several long-standing conjectures (the virtual Haken conjecture, the virtual fibering conjecture, and the LERF conjecture for hyperbolic 3-manifold groups). The proof relies crucially on the theory of CAT(0) cube complexes and special groups developed by Wise, together with Kahn and Markovic’s construction of essential surfaces via almost-geodesic immersed surfaces.

Remark 7.24. The resolution of the virtual Haken conjecture beautifully illustrates the power of geometric group theory: a topological question about 3-manifolds was resolved using the geometric theory of CAT(0) cube complexes, the combinatorial theory of special groups and RAAGs, the ergodic-theoretic construction of surfaces by Kahn-Markovic, and the algebraic theory of quotients of hyperbolic groups developed by Wise. This synthesis of techniques from across mathematics is characteristic of modern geometric group theory.

7.10 Open Problems

We conclude with a selection of major open problems that continue to drive research in geometric group theory.

Open Problem 1 (Cannon Conjecture). If \(G\) is a hyperbolic group with \(\partial_\infty G \cong S^2\), is \(G\) the fundamental group of a closed hyperbolic 3-manifold?

Open Problem 2 (Flat Closing Conjecture). If a CAT(0) group \(G\) contains \(\mathbb{Z}^n\) but not \(\mathbb{Z}^{n+1}\), does the CAT(0) space on which \(G\) acts contain an isometrically embedded copy of \(\mathbb{R}^n\)?

Open Problem 3 (Quasi-Isometric Classification of 3-Manifold Groups). Complete the quasi-isometric classification of fundamental groups of compact 3-manifolds.

Open Problem 4 (Linearity of Hyperbolic Groups). Is every hyperbolic group linear (i.e., isomorphic to a subgroup of \(\mathrm{GL}(n, \mathbb{C})\) for some \(n\))?

Open Problem 5 (CAT(0) Groups and the Word Problem). Does every CAT(0) group have solvable conjugacy problem? (The word problem is known to be solvable.)

Open Problem 6 (Residual Finiteness of Hyperbolic Groups). Is every hyperbolic group residually finite?

These problems reflect the vitality and depth of geometric group theory as a field. From its origins in the work of Dehn on surface groups and the combinatorial methods of the early 20th century, through the revolutionary insights of Gromov, Thurston, and their contemporaries, to the modern synthesis involving cube complexes, boundaries, and acylindrical actions, geometric group theory continues to reveal profound connections between algebra, geometry, and topology.