Editing Geometric group theory (section)

{{Short description|Area in mathematics devoted to the study of finitely generated groups}}
[[File:F2 Cayley Graph.png|thumb|The [[Cayley graph]] of a [[free group]] with two generators. This is a [[hyperbolic group]] whose [[Gromov boundary]] is a [[Cantor set]]. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.]]

'''Geometric group theory''' is an area in [[mathematics]] devoted to the study of [[finitely generated group]]s via exploring the connections between [[algebra]]ic properties of such [[group (mathematics)|groups]] and [[topology|topological]] and [[geometry|geometric]] properties of spaces on which these groups can [[Group action (mathematics)|act]] non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).

Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the [[Cayley graph]]s of groups, which, in addition to the [[graph (discrete mathematics)|graph]] structure, are endowed with the structure of a [[metric space]], given by the so-called [[word metric]].

Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with [[low-dimensional topology]], [[hyperbolic geometry]], [[algebraic topology]], [[computational group theory]] and [[differential geometry]]. There are also substantial connections with [[computational complexity theory|complexity theory]], [[mathematical logic]], the study of [[Lie group]]s and their discrete subgroups, [[dynamical systems]], [[probability theory]], [[K-theory]], and other areas of mathematics.

In the introduction to his book ''Topics in Geometric Group Theory'', [[Pierre de la Harpe]] wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things that [[Georges de Rham]] practiced on many occasions, such as teaching mathematics, reciting [[Stéphane Mallarmé|Mallarmé]], or greeting a friend".<ref>P. de la Harpe, [https://books.google.com/books?id=60fTzwfqeQIC&dq=de+la+Harpe,+Topics+in+geometric+group+theory&pg=PP1 ''Topics in geometric group theory''.] Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. {{ISBN|0-226-31719-6}}, {{ISBN|0-226-31721-8}}.</ref>{{rp|3}}