Breadcrumb
Geometric Group Theory
Geometric group theory is the study of geometric properties of infinite discrete groups.
The field is relatively young but has roots in older areas, most prominently low-dimensional topology and algebraic topology. In the former, combinatorial techniques were developed to understand knots, surfaces and 3-manifolds via associated groups. In the latter, algebraic theories (particularly cohomology) were used to distinguish amongst spaces and amongst groups.
In the 1980s, spurred by ideas of J.W. Cannon and M. Gromov, researchers began to pay attention to geometric structures associated with infinite groups. This shed much light on the earlier combinatorial and topological investigations and stimulated innovations which have since developed rapidly.
As it has grown, geometric group theory has interacted with many disciplines including Riemannian geometry, coarse geometry, algebraic topology, low-dimensional topology, analysis, dynamical systems, ergodic theory, combinatorics, computer science and logic.
The subject is often motivated by or focuses on a number of rich examples such as Artin groups, Coxeter groups, braid groups, mapping class groups, 3-manifold groups, lattices in Lie groups, nilpotent groups, polycyclic groups, solvable groups, hyperbolic groups, CAT(0) groups, Thompson's group, automata groups and many others.
Bristol is part of a research network Geometric and Analytic Methods in Group Theory together with Oxford and Southampton Universities.
A public domain image created using software of J. Weeks.
A dodecahedral hyperbolic tessellation arising from a discrete group acting on 3-dimensional hyperbolic space.