Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (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 graphs of groups, which, in addition to the 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 has become a clearly identifiable branch of mathematics in 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 complexity theory, mathematical logic, the study of Lie Groups 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 practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend" (page 3 in ).
Other articles related to "geometric group theory, groups, group, theory":
... For infinite groups, the coarse geometry of the Cayley graph is fundamental to geometric group theory ... For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group ... This is only interesting for infinite groups every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group ...
... A presentation of a group determines a geometry, in the sense of geometric group theory one has the Cayley graph, which has a metric, called the word metric ... An important example is in the Coxeter groups ...
... In mathematics, given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h G → H such that for all u and v in G it ... to inverses in the sense that Hence one can say that h "is compatible with the group structure" ... A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h ...
... Cannon was one of the co-authors of the 1992 book "Word Processing in Groups" which introduced, formalized and developed the theory of automatic groups ... The theory of automatic groups brought new computational ideas from computer science to geometric group theory and played an important role in the development of the subject in 1990s ... The goal was to understand when an action of a group by homeomorphisms on a 2-sphere is (up to a topological conjugation) an action on the standard Riemann sphere by Möbius ...
... 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 ... Wallpaper groups Baumslag-Solitar groups Fundamental groups of graphs of groups Grigorchuk group ...
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