On the existence of representations of finitely presented groups in compact Lie groups

TL;DR

This paper investigates conditions under which the fundamental group of certain 2-complexes can be represented in compact Lie groups, focusing on cases with specific topological properties and prescribed loop values.

Contribution

It establishes new existence results for representations of finitely presented groups into compact Lie groups under topological and cohomological constraints.

Findings

Existence of representations when $b_2(X)=0$ or $b_2(X)=1$ with specific conditions

Representation existence in $SO(3)$ when cup product is non-zero

Results depend on the second Betti number and cohomological properties

Abstract

Given a finite, connected 2-complex $X$ such that $b_2(X)\le1$ we establish two existence results for representations of the fundamental group of $X$ into compact connected Lie groups $G$, with prescribed values on certain loops. If $b_2(X)=1$ we assume $G=SO(3)$ and that the cup product on the first rational cohomology group of $X$ is non-zero.