Representation spaces for central extensions and almost commuting unitary matrices

TL;DR

This paper studies the topological structure of spaces of homomorphisms from certain central extensions of abelian groups into unitary groups, revealing insights into their representation spaces and moduli.

Contribution

It provides a detailed description of the topology of homomorphism and representation spaces for specific central extensions into unitary groups, a topic not extensively explored before.

Findings

Topology of Hom(mma, U(m)) characterized

Structure of associated moduli spaces analyzed

Insights into almost commuting unitary matrices obtained

Abstract

Let $\Gamma$ denote a central extension of the form $1\to \mathbb{Z}^r\to\Gamma\to \mathbb{Z}^n\to 1$. In this paper we describe the topology of the spaces of homomorphisms $\text{Hom}(\Gamma, U(m))$ and the associated moduli spaces $\text{Rep}(\Gamma, U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.