Geometric group theory

Summary

Geometric group theory studies finitely generated groups through the geometry of spaces on which they act, such as their Cayley graphs with the word metric. Its basic principle is to study groups up to quasi-isometry, which ignores small-scale details and distortions by bounded factors. The Švarc–Milnor lemma links this view to group actions on metric spaces. Central results include Gromov's theorem on groups of polynomial growth and Stallings's theorem on ends of groups. Central classes include hyperbolic groups and amenable groups. 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.