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In this paper the space of commuting elements in the central product$G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where$p$ is a prime number. In particular, a computation for the number of pathconnected components of these spaces is given and the geometry of the modulispace $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles overthe $n$--torus is completely described for all values of $n$, $m$ and $p$.

Commuting elements in central products of special unitary groups