Browsing by Author "Stafa, Mentor"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Hilbert–Poincaré series for spaces of commuting elements in Lie groups
    (Springer, 2018) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of Science
    In this article we study the homology of spaces Hom(Zn,G) of ordered pairwise commuting n-tuples in a Lie group G. We give an explicit formula for the Poincaré 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 Hom(Fn/Γmn,G) , where the subgroups Γmn are the terms in the descending central series of the free group Fn . Finally, we show that there is a stable equivalence between the space Comm(G) studied by Cohen–Stafa and its nilpotent analogues.
  • Thumbnail Image
    Item
    Homological Stability for Spaces of Commuting Elements in Lie Groups
    (Oxford, 2021-03) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of Science
    In this paper, we study homological stability for spaces Hom(Zn,G) of pairwise commuting n-tuples in a Lie group G⁠. We prove that for each n⩾1⁠, these spaces satisfy rational homological stability as G ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, Comm(G) and BcomG⁠, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases. Our proofs use the theory of representation stability—in particular, the theory of FIW-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.
  • Thumbnail Image
    Item
    Polyhedral products, flag complexes and monodromy representations
    (Elsevier, 2018-08) Stafa, Mentor; Mathematical Sciences, School of Science
    This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups and outer automorphisms of free groups , respectively, as well as faithful representations of products of finite groups into the linear groups and . These faithful representations are realized as monodromy representations.