Jason Bode

Ph.D. in Mathematics (May 2007)

First Position

Assistant professor at Lander University

Dissertation

Isoperimetric Constants and Self-Avoiding Walks and Polygons on Hyperbolic Coxeter Groups

Advisor

Kenneth S. Brown

Research Area

Topology, algebra, self-avoiding walks on Cayley graphs

Abstract

We study isoperimetric constants of and self-avoiding walks (SAWs) and self-avoiding polygons (SAPs) on Cayley graphs of hyperbolic Coxeter groups. These graphs are tilings of the hyperbolic plane. We calculate the isoperimetric constants of most rank three hyperbolic Coxeter groups and give an estimate for the remainder. We prove that for all but a few classes of these groups there are exponentially fewer k-step SAPs than there are k-step SAWs. We prove that this is also true of any nontrivial free product. Additionally we show that there are exponentially more k-step SAWs on a free group F_n (resp. a “free” Coxeter group T_d) than on any strict subgroup of F_n (resp. T_d).