Hilbert-Poincaré series for spaces of commuting elements in Lie groups (Q2633152)

Let $G$ be a compact and connected Lie group $G$ and $\pi$ a discrete group $\pi$ generated by $n$ elements. The authors consider the rational homology of the space of group homomorphisms $\Hom(\pi, G)\subseteq G^n$, endowed with the subspace topology from $G^n$ and give an explicit formula for the Poincaré series of $\Hom(\pi, G)_1$, the connected component of the trivial representation, when $\pi$ is free abelian or nilpotent. \par The formula given for the Poincaré series of the identity component $\Hom(\mathbb{Z}^n, G)_1$ is based on the works [\textit{T. J. Baird}, Algebr. Geom. Topol. 7, 737--754 (2007; Zbl 1163.57026)] and [\textit{F. R. Cohen} and \textit{M. Stafa}, Math. Proc. Camb. Philos. Soc. 161, No. 3, 381--407 (2016; Zbl 1371.55012)].