Gromov Hyperbolic Groups Essays In Group Theory.
James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University. James W. Cannon. Born January 30, 1943 (age 77) Bellefonte, Pennsylvania. Nationality: American: Citizenship: United States: Alma mater: Ph.D. (1969), University of Utah.
In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion.
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group.The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.
We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the K-homology of the corona with the K-theory of the Roe algebra, via the coarse assembly map.We also establish a dual theory, that is, we relate the K-theory of the corona with the K-theory of the reduced stable Higson corona via the coarse co-assembly map.
Geometric ideas have played a very major role in the development of Group theory in 20th century. Coxeter groups and Coxeter complexes, Tits theory of buildings, Bass-Serre theory of Groups acting on trees, Gromov’s notion of hyperbolic groups and combinatorial methods have all been very central in several areas of mathe-matics.
The purpose of this article is then to generalize the first two items in Examples 1.2 to the setting of relatively hyperbolic groups. Recently, the first-named author has established in his thesis ( 15 ) that relatively hyperbolic groups are statistically hyperbolic, provided that the group growth rate dominates the ones of parabolic subgroups.
Accessibility is concerned with bounding the complexity of decompositions as graphs of groups for a discrete group. In his proof that groups of cohomological dimension one are necessarily free, J.R. Stallings made use of Grushko's theorem, which asserts that if a group is generated by a subset of cardinality and decomposes as a non-trivial free product, then.