Geometric Group Theory — Mathematics Knowledge System

Overview

Geometric group theory studies groups by equipping them with geometric structure and analyzing their large-scale (coarse) geometry. The central idea is that a finitely generated group has a well-defined large-scale geometry given by its Cayley graph, and groups that are quasi-isometric share many algebraic properties. The field was transformed by Gromov's introduction of hyperbolic groups in 1987, connecting group theory to differential geometry and dynamics.

Intuition

Geometric group theory says: to understand a group, make it act on a nice geometric space and study the geometry. A finitely generated group G with generating set S has a Cayley graph whose metric gives G a geometry. Groups that look the same at large scales (quasi-isometric) share many properties — polynomial growth, hyperbolicity, number of ends. Hyperbolic groups, which act on negatively curved spaces, have solvable word problems and many nice properties.

Formal Definition

Definition

The Cayley graph and quasi-isometry are the fundamental constructions. The word metric makes G into a metric space.

d_S(g, h) = \min\{n : g^{-1}h = s_1 \cdots s_n,; s_i \in S \cup S^{-1}\}

Word metric on G with generating set S

f : X \to Y \text{ is a quasi-isometry if } \exists\, C, K > 0:; \frac{1}{C}d_X(x,x') - K \leq d_Y(f(x),f(x')) \leq C d_X(x,x') + K

Quasi-isometry

\delta\text{-hyperbolic: every geodesic triangle is }\delta\text{-slim (each side lies in }\delta\text{-neighborhood of other two)}

Gromov hyperbolicity

\beta_G(n) = |\{g \in G : d_S(e, g) \leq n\}|

Growth function of G

Notation

Notation	Meaning
$\mathrm{Cay}(G, S)$	Cayley graph of group G with generating set S
$d_S$	Word metric on G with respect to S
$\simeq_{\mathrm{QI}}$	Quasi-isometry equivalence
$\beta_G(n)$	Growth function: number of group elements within distance n

Theorems

Theorem 1: Gromov's Polynomial Growth Theorem

A finitely generated group has polynomial growth if and only if it is virtually nilpotent

Theorem 2: Milnor-Schwarz Lemma

If a group G acts properly and cocompactly by isometries on a geodesic metric space X, then G is finitely generated and quasi-isometric to X

Theorem 3: Stallings' Ends Theorem

A finitely generated group G has more than one end if and only if G splits as an amalgam or HNN-extension over a finite subgroup

Theorem 4: Dehn's Algorithm for Hyperbolic Groups

Every hyperbolic group has a solvable word problem; Dehn's algorithm terminates in linear time

Worked Examples

1
Vertices are elements (m,n) of Z^2; edges connect (m,n) to (m+1,n) and (m,n+1) (and their inverses).
$\mathrm{Cay}(\mathbb{Z}^2, \{(1,0),(0,1)\})$
2
The Cayley graph is the integer lattice grid in the plane. The word metric gives taxicab (L^1) distance.
$d_S((m,n),(m',n')) = |m-m'| + |n-n'|$
3
At large scale, this looks like the Euclidean plane R^2, confirming Z^2 is quasi-isometric to R^2.
$\mathbb{Z}^2 \simeq_{\mathrm{QI}} \mathbb{R}^2$

✓ Answer

The Cayley graph of Z^2 is the integer lattice, quasi-isometric to the Euclidean plane.

Practice Problems

Hardproof writing

Prove that the word metric on a finitely generated group G is well-defined up to quasi-isometry, i.e., changing the generating set gives a quasi-isometric metric.

Hardfree response

What does it mean for a group to be hyperbolic in the sense of Gromov, and give two equivalent characterizations.

Common Mistakes

Common Mistake

Thinking quasi-isometry is the same as homeomorphism

Quasi-isometry is a large-scale notion that ignores small-scale structure. Homeomorphism is a fine-scale topological notion. R and R^2 are not quasi-isometric (different growth) but many non-homeomorphic spaces can be quasi-isometric.

Common Mistake

Assuming the word metric depends on the choice of generating set

Different finite generating sets give different word metrics, but they are all quasi-isometric to each other. The large-scale geometry is an intrinsic property of the group.

Quiz

Historical Background

The roots lie in Poincare's use of group actions to study surfaces and Dehn's word and conjugacy problems (1911). The modern field crystallized with the work of Stallings on ends of groups and Gromov's 1981 theorem that polynomial growth implies virtually nilpotent. Gromov's 1987 monograph on hyperbolic groups launched the contemporary era of geometric group theory.

1911
Dehn formulates the word problem, conjugacy problem, and isomorphism problem for groups
Max Dehn
1968
Stallings proves groups with more than one end are amalgams
John Stallings
1981
Gromov proves groups of polynomial growth are virtually nilpotent
Mikhail Gromov
1987
Gromov introduces hyperbolic groups and quasi-isometry
Mikhail Gromov
1991
Bridson-Haefliger develop CAT(0) space theory for group actions
Martin Bridson, Andre Haefliger

Summary

Geometric group theory studies groups via their actions on metric spaces and their large-scale geometry.
The Cayley graph equips a finitely generated group with a metric (the word metric).
Quasi-isometry is the key equivalence relation: groups with the same large-scale geometry share many properties.
Gromov's polynomial growth theorem: polynomial growth equivalent to virtually nilpotent.
Hyperbolic groups (Gromov hyperbolicity) have solvable word problems and rich geometric properties.

References

BookBridson, M.R., Haefliger, A. — Metric Spaces of Non-Positive Curvature (1999), Springer
WebsiteWikipedia — Geometric group theory
WebsitenLab — geometric group theory