---
tag: 2T3218RXTD
kind: concept
name: Geometric group theory
msc: ["20F65"]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_group_theory
same_as:
  wikidata: Q2662474
  wikipedia: Geometric group theory
---

## Summary

Geometric group theory studies [[finitely generated groups|082DWWXMRJ]] through the geometry of spaces on which they act, such as their [[Cayley graphs|0GYQK9NDJF]] with the [[word metric|VCHJVR9NGH]].
Its basic principle is to study groups up to [[quasi-isometry|6FKYSH729T]], which ignores small-scale details and distortions by bounded factors.
The [[Švarc–Milnor lemma|NTK1TVA50A]] links this view to group actions on metric spaces.
Central results include [[Gromov's theorem on groups of polynomial growth|RSBFHHPTWX]] and [[Stallings's theorem on ends of groups|GWZN3EKEWD]].
Central classes include [[hyperbolic groups|JWJ8YPR7EB]] and [[amenable groups|CEAT9BESKK]].
The field grew from combinatorial group theory and low-dimensional topology.
It took its modern form in the 1980s and 1990s, largely through the work of Gromov.

## Explanations

### L1 human

A group given by generators can be drawn as a graph: one dot for each element, and a line from $g$ to $gs$ for each generator $s$.
Seen from far away, this graph has a shape.
The Cayley graph of $\Z^2$ with its two standard generators looks like a grid and, from far away, like a plane.
The Cayley graph of a free group on two generators is an infinite tree, which looks nothing like a plane.
Geometric group theory reads algebraic properties of the group from such large-scale shapes, and the other way around.