of groups whose amenability status is still unknown. Interactions with finite group theory, particularly progress in the study of subgroup growth. Studying subgroups and lattices in linear groups, such as , and of other Lie groups, via geometric methods (e.g. buildings), algebro-geometric tools (e.g. algebraic groups and representation varieties), analytic methods (e.g. unitary representations on Hilbert spaces) and arithmetic methods. Group cohomology, using algebraic and topological methods, particularly involving interaction with algebraic topology and the use of morse-theoretic ideas in the combinatorial context; large-scale, or coarse (see e.g.) homological and cohomological methods. Progress on traditional combinatorial group theory topics, such as the Burnside problem, the study of Coxeter groups and Artin groups, and so on (the methods used to study these questions currently are often geometric and topological). Examples The following examples are often studied in geometric group theory: Amenable groups Free Burnside groups 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) Symmetric groups Braid groups Coxeter groups General Artin groups Thompson's group F CAT(0) groups Arithmetic groups Automatic groups Fuchsian groups, Kleinian groups, and other groups acting properly discontinuously on symmetric spaces, in particular lattices in semisimple Lie groups. Wallpaper groups Baumslag–Solitar groups Fundamental groups of graphs of groups Grigorchuk group See also The ping-pong lemma, a useful way to exhibit a group as a free product Amenable group Nielsen transformation Tietze transformation References Books and monographs These texts cover geometric group theory and related topics. External links Jon McCammond's Geometric Group Theory Page What is Geometric Group Theory? By Daniel Wise Open Problems in combinatorial and geometric group theory Geometric group theory Theme on arxiv.org Group theory