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Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
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 .
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 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.
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups.
In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space, is quasi-isometric to .
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of