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In this article we study the homology of spaces ${rm Hom}(mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${rm Hom}(F_n/Gamma_n^m,G)$, where the subgroups $Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.
In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that the
By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in the sense
By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesins entropy formula and SRB measures of a finitely generated random transformations on
We define and study non-abelian Poincare series for the group $G=mathrm{SU} (2,1)$, i.e. Poincare series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincare series have in general exponential growth. In