Geometric group theory - Revision history

Karl Jones at 14:57, 22 September 2016

2016-09-22T14:57:25Z

Karl Jones at 14:49, 22 September 2016

2016-09-22T14:49:53Z

Karl Jones: Created page with "'''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..."

2016-09-22T14:47:39Z

Created page with "'''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..."

New page

'''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).

== Description ==

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 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 [[complexity theory]], [[mathematical logic]], the study of [[Lie group|Lie Groups]] and their discrete subgroups, [[Dynamical system|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".

== See also ==

The [[ping-pong lemma]], a useful way to exhibit a group as a free product
* [[Amenable group]]
* [[Nielsen transformation]]
* [[Tietze transformation]]

== External links ==

* [https://en.wikipedia.org/wiki/Geometric_group_theory Geometric group theory] @ Wikipedia

[[Category:Mathematics]]

← Older revision		Revision as of 14:57, 22 September 2016
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−	'''Geometric group theory''' is an area in [[mathematics]] devoted to the study of [[Finitely generated group\|finitely generated groups]] via exploring the connections between algebraic properties of such groups and [[Topology\|topological]] and [{Geometry\|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).	+	'''Geometric group theory''' is an area in [[mathematics]] devoted to the study of [[Finitely generated group\|finitely generated groups]] via exploring the connections between algebraic properties of such groups and [[Topology\|topological]] and [[Geometry\|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).

	== Description ==		== Description ==

@@ Line 1: / Line 1: @@
-'''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).
+'''Geometric group theory''' is an area in [[mathematics]] devoted to the study of [[Finitely generated group|finitely generated groups]] via exploring the connections between algebraic properties of such groups and [[Topology|topological]] and [{Geometry|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).
 == Description ==
-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.
+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 became a clearly identifiable branch of mathematics in the late 1980s and early 1990s.
@@ Line 15: / Line 15: @@
 == See also ==
 * [[Amenable group]]
 * [[Nielsen transformation]]
 * [[Tietze transformation]]