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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 can act non-trivially (that is, when the groups in question are realized as geometric symmetries or ...
In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions: Each element of G acts as an isometry of X. The action is cocompact, i.e. the quotient space X/G is a compact space.
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. [7] The first idea is made precise by means of the Cayley graph , whose vertices correspond to group elements and edges correspond to right multiplication in the 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. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
In mathematics, geometric group theory is the study of groups by geometric methods. See also Category:Combinatorial group theory . The main article for this category is Geometric group theory .
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, [1] characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees.It was introduced in unpublished work of Eliyahu Rips in about 1991.. An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval.
In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. [h] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. [40] Further branches crucially applying groups include algebraic geometry and number ...