**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 ).

Read more about Geometric Group Theory: History, Modern Themes and Developments, Examples

### Other articles related to "geometric group theory, groups, group, theory":

**Geometric 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**...

**Geometric Group Theory**

... 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 ...

**Geometric Group Theory**- 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 ... Wallpaper

**groups**Baumslag-Solitar

**groups**Fundamental

**groups**of graphs of

**groups**Grigorchuk

**group**...

### Famous quotes containing the words theory, geometric and/or group:

“The *theory* seems to be that so long as a man is a failure he is one of God’s chillun, but that as soon as he has any luck he owes it to the Devil.”

—H.L. (Henry Lewis)

“New York ... is a city of *geometric* heights, a petrified desert of grids and lattices, an inferno of greenish abstraction under a flat sky, a real Metropolis from which man is absent by his very accumulation.”

—Roland Barthes (1915–1980)

“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another *group* is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”

—Gore Vidal (b. 1925)