Trace Preserving Homomorphisms on SL(2,C)

TL;DR

This paper characterizes trace preserving homomorphisms on subgroups of SL(2,C), especially Fuchsian groups with elliptic elements, and determines how traces of generators define the group up to conjugation.

Contribution

It extends existing results to include Fuchsian groups with elliptic elements and provides elementary methods to identify the minimal trace data needed for group classification.

Findings

Characterizes when trace invariant homomorphisms are conjugations in SL(2,R)

Determines trace conditions for finitely presented groups

Expands classification results to Fuchsian groups with elliptic elements

Abstract

Let G a be subgroup of SL(2,C), the group of 2x2 matrices of determinant 1 with complex entries. Let h map onto h(G) be a homomorphism. We call h a trace preserving homomorphism if tr(h(g))=tr(g) for all g in G,where tr(g) is the trace of g. We solve the question of when a trace invariant homomorphism is a conjugation by some A in SL(2,R). Moreover, if the group G is finitely presented, this paper determines which traces of the generators and products of the generators determine the group up to conjugation. Incomplete solutions are known from the study of Fuchsian groups. Our theorems in this paper will expand the results in the literature to include Fuchsian Groups with elliptic elements, which have not been considered before. Moreover, they will be applicable to any class of subgroups of SL(2,C). The methods used will be relatively elementary and will indicate how many traces are needed, and the role that any relator equation plays in the parameterization by traces of a class of groups.