A Presentation of Two Families of Uniformly Bounded Representations of CAT(0)-Cubical Groups and an Example from Hyperbolic Geometry

Abstract

Geometric group theory is a branch of mathematics in which we explore the characteristics of finitely-generated groups by letting the group act on a particular space and by analyzing the connections between the group’s algebraic properties and the geometric and topological properties of the spaces being acted upon. In the last half of the 20th century, harmonic analysis on a free group was extensively studied and Hilbert space representations of the free group were an integral tool in this research. In 1986, T. Pytlik and R. Szwarc [15] constructed a particularly useful family of uniformly bounded representations of the free group F acting (by translation) on `2(F). In this dissertation we will extend Pytlik and Szwarc’s construction of a holomorphic family of uniformly bounded Hilbert space representations for the free group F acting on `2(F) to the more general case of a discrete group acting on `2(X), where X is the set of vertices of a CAT(0)-cube complex. We will then show that these representations are identical to another holomorphic family of uniformly bounded Hilbert space representations constructed by E. Guentner and N. Higson using cocycles. We also examine an example of a discrete group acting on a non-positively curved cube-complex which yields the result that, for every 3-manifold group, there exists a non-positively curved space on which it acts freely.