Topics in algebra :the Higman-Thompson group G b2 s, b1 s and Beauville p-groups

Abstract

This thesis consists of two parts. Part I of this thesis is concerned with the Higman-Thompson group G2,1. We review and apply Definitions, Lemmas and Theorems described in a series of lectures delivered by Graham Higman during a visit to the Australian National University from July 1973 to October 1973 on a family of finitely presented infinite groups Gn,r for n 2 and r 1. This thesis will concentrate on the group G2,1 (otherwise know as Thompson’s group V). We give a brief account of the history of the Higman-Thompson group G2,1, we clarify the proof of the conjugacy problem for elements in quasi-normal form and we prove that the power conjugacy problem for the group G2,1 is decidable. Part II of this thesis concentrates on the existence and structure of mixed and unmixed Beauville p-groups, for p a prime. We start by exhibiting the first explicit family of mixed Beauville 2-groups and find the corresponding surfaces. We follow this up by exploring the method that was used to construct the family; this leads to further ramification structures for finite p-groups giving rise to surfaces isogenous to a higher product of curves. We finish by classifying the non-abelian Beauville pgroups of order p3, p4 and provide partial results for p-groups of order p5 and p6. We also construct the smallest Beauville p-groups for each prime p.