Representation spaces for central extensions and almost commuting unitary matrices

Let $\mathrm{\Gamma }$ 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 $\text{Hom}(\mathrm{\Gamma },U(m))$ and the associated moduli spaces $\text{Rep}(\mathrm{\Gamma },U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.