Related papers: Canonical representations of surface groups

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation…

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider $(2g+1)$-dimensional complex linear representations of the pure mapping class groups of…

Let $\phi \in {\rm Mod}(\Sigma)$ be an arbitrary element of the mapping class group of a closed orientable surface $\Sigma$ of genus at least $2$. For any characteristic cover $\widetilde{\Sigma} \to \Sigma$ one can consider the linear…

Log-Riemann surfaces of finite type are Riemann surfaces with finitely generated fundamental group equipped with a local diffeomorphism to C such that the surface has finitely many infinite order ramification points. We define and prove…

Given a finitely-generated group $\pi$ and a linear algebraic group $G$, the representation variety Hom$(\pi,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $G$. Its algebraic…

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of…

We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if $f\colon X\rightarrow \mathrm{Spec}(k)$ is a projective scheme over a field $k$ and $G$ is a finite…

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be…

We consider the vector space $E_{\rho,p}$ of entire functions of finite order, whose types are not more than $p>0$, endowed with Frechet topology, which is generated by a sequence of weighted norms. We call a function $f\in E_{\rho,p}$ {\it…

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura…

The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of…

In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let $\Sigma_{g}$ denote a…

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than…

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental group…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…

We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(\pi_1(\Sigma),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$…

Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of…

For a finite group $G$, there is a map $RO(G) \to {\rm Pic}(Sp^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map…

Lyubashenko's construction associates representations of mapping class groups Map_{g,n} of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the…