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 ...
Gruppentheorie und Quantenmechanik, or The Theory of Groups and Quantum Mechanics, is a textbook written by Hermann Weyl about the mathematical study of symmetry, group theory, and how to apply it to quantum physics.
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 the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram [1] [2] [3]) is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that 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 .
Geometric group theory is a relatively new area of mathematics beginning around the late 1980s [1] which explores finitely generated groups, and seeks connections between their algebraic properties and the geometric spaces on which these groups act.
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
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