arXiv subjects

Mentor Stafa

Publications and source records attributed to Mentor Stafa.

8 recordsLinked to original sources

arXiv2020

Homological stability for spaces of commuting elements in Lie groups

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 $n\geqslant 1$, 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, ${\rm Comm}(G)$ and ${\rm B_{com}} G$, 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 ${\rm FI}_W$-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.

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arXiv2018

Hilbert-Poincare series for spaces of commuting elements in Lie groups

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 ${\rm Hom}(F_n/Γ_n^m,G)$, where the subgroups $Γ_n^m$ 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.

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arXiv2018

Polyhedral products, flag complexes and monodromy representations

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 $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations.

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arXiv2017

Poincare series of character varieties for nilpotent groups

For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $Γ$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(Γ,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to the order of the Weyl group $W$. We give explicit formulas for the Poincaré series. In addition we study $G$-equivariant stable decompositions of subspaces $X(q,G)$ of the free monoid $J(G)$ generated by the Lie group $G$, obtained from finitely generated free nilpotent group representations.

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arXiv2016

A survey on spaces of homomorphisms to Lie groups

The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups $G$, and to describe their connections to classical representation theory, as well as other structures. Various properties are given when $G$ is replaced by a small category, or the discrete group is given by a right-angled Artin group.

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arXiv2016

On spaces of commuting elements in Lie groups

The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the Weyl group inverted.

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arXiv2015

On the fundamental group of certain polyhedral products

Let $K$ be a finite simplicial complex, and $(X,A)$ be a pair of spaces. The purpose of this article is to study the fundamental group of the polyhedral product denoted $Z_K(X,A)$, which denotes the moment-angle complex of Buchstaber-Panov in the case $(X,A) = (D^2, S^1)$, with extension to arbitrary pairs in [2] as given in Definition 2.2 here. For the case of a discrete group $G$, we give necessary and sufficient conditions on the abstract simplicial complex $K$ such that the polyhedral product denoted by $Z_K(\underline{BG})$ is an Eilenberg-Mac Lane space. The fundamental group of $Z_K(\underline{BG})$ is shown to depend only on the 1-skeleton of $K$. Further special examples of polyhedral products are also investigated. Finally, we use polyhedral products to study an extension problem related to transitively commutative groups, which are given in Definition 5.2.

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arXiv2015

On monodromy representations in Denham-Suciu fibrations

We study the monodromy representation corresponding to a fibration introduced by G. Denham and A. Suciu, which involves polyhedral products given in Definition 2.2. Algebraic and geometric descriptions for these monodromy representations are given. In particular, we study the case of a product of two finite cyclic groups and obtain representations into $Out(F_n)$ and $SL_n(\mathbb Z)$. We give algebraic descriptions of monodromy for the case of a product of any two finite groups . Finally we give a geometric description for monodromy representations of a product of 2 or more finite groups to $Out(F_n)$, as well as some algebraic properties. The geometric description does not rely on choosing a basis for the fundamental group of the fibre in terms of commutators, hence avoids this delicate question.

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