Geometric group theory
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Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups.
Geometric group theory uses topological and geometric methods to study groups; the main philosophy is to deduce information about a group by analyzing how it acts on topological spaces. Combinatorial group theory studies discrete groups as quotients of free groups, typically described using presentations. In the early 20th century, pioneering work of Dehn, Nielsen, Reidemeister and Schreier amongst others established a close correspondence between the two subjects. While some problems and methods are still discernibly "more geometric" or "more combinatorial" than others, the fields are inextricably intertwined; they are now generally considered the same area of mathematics. Other closely related fields include algebraic topology, geometric topology and computational group theory.
Since the early 1980's there have developed important broad themes which motivate the study of arbitrary finitely generated groups. Particularly influential is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry. This program involves:
1) A description of properties that are invariant under quasi-isometry, for example
- the growth rate of a finitely generated group
- the isoperimetric function or Dehn function of a finitely generated group
- ends of a group
- hyperbolicity of a group, and the boundary of a hyperbolic group
2) Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example
- Stallings ends theorem
3) Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space.
[edit] Examples
The following examples are often studied in geometric group theory:
- The infinite cyclic group Z
- Free groups
- Free products
- Outer automorphism groups Out(Fn) (via Outer space)
- Hyperbolic groups
- Mapping class groups (automorphisms of surfaces)
- Braid groups
- General Artin groups
- Thompson's group F
- CAT(0) groups
- Arithmetic groups
- Automatic groups
- Kleinian groups, and other lattices acting on symmetric spaces.
- Wallpaper groups
[edit] See also
- The Cayley graph, the "canonical" choice of space for a group action
- The Ping-Pong lemma, a useful way to exhibit a group as a free product
- Amenability
