The description of the space of commuting elements in a compact Lie group is an interesting algebro-geometric problem with applications in Mathematical Physics, notably in Supersymmetric Yang Mills theories. When the Lie group is complex reductive, this space is the character variety of a free abelian group. Let $K$ be a compact Lie group (not necessarily connected) and $G$ be its complexification. We consider, more generally, an arbitrary finitely generated abelian group $A$, and show that the conjugation orbit space Hom$(A,K)/K$ is a strong deformation retract of the character variety Hom$(A,G)/G$ (the same result was also shown recently for a nilpotent group replacing $A$). As a Corollary, in the case when $G$ is connected and semisimple, we obtain necessary and sufficient conditions for Hom$(A,G)/G$ to be irreducible. This relates to $G$-Higgs bundles over abelian varieties, to interesting problems on irreducibility of the variety of $k$ tuples of $n$ by $n$ commuting matrices, and to the Hilbert scheme of $n$ points on $\mathbb C^k$.