Search results
Results from the WOW.Com Content Network
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 ...
The Rips machine assigns to a stable isometric action of a finitely generated group G a certain "normal form" approximation of that action by a stable action of G on a simplicial tree and hence a splitting of G in the sense of Bass–Serre theory. Group actions on real trees arise naturally in several contexts in geometric topology: for example ...
In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations.
An earlier result of Joseph A. Wolf [2] showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h [3] and independently Hyman Bass [4] (with different proofs) computed the exact order of polynomial growth. Let G be a finitely generated nilpotent group with lower central series
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.
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 .
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.
Karen Vogtmann FRS (born July 13, 1949 in Pittsburg, California [1]) is an American mathematician working primarily in the area of geometric group theory.She is known for having introduced, in a 1986 paper with Marc Culler, [2] an object now known as the Culler–Vogtmann Outer space.