Related papers: Algebraic Mapping Class Group Rigidity

We establish the existence of two rigid triples of conjugacy classes in the algebraic group $G_{2}$ in characteristic $5$, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups $G_{2}(5^n)$…

We compute the automorphism group of the intersection graph of many large-type Artin groups. This graph is an analogue of the curve graph of mapping class groups but in the context of Artin groups. As an application, we deduce a number of…

We find a sharp bound for the order of the automorphism group of a stable curve of genus $g$ with $3g-3$ nodes, and a sharp bound for the order of the automorphism group of such a curve with all smooth components. Combined with the results…

We classify the pairs $(X,\pi)$, where $\pi\colon X\to S$ is a $\mathbb{P}^1$-bundle over a non-rational geometrically ruled surface $S$ and $\mathrm{Aut}^\circ(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the…

Let $\Gamma$ be a finite index subgroup of the mapping class group $MCG(\Sigma)$ of a closed orientable surface $\Sigma$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element…

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$.…

Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For…

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping…

The ${\rm SL}_3$-skein algebra of a closed surface $\Sigma_g$ is a quantization of the ${\rm SL}_3$ character variety of $\Sigma_g$, where $g$ denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web…

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product…

Let $N_{g,n}$ be a genus $g$ compact non-orientable surface with $n$ boundaries. We explain about relations on the level $d$ mapping class group $\mathcal{M}_d(N_{g,0})$ of $N_{g,0}$ and the level $d$ principal congruence subgroup…

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

The point-pushing subgroup P(S) of the mapping class group Mod(S) of a surface with marked point is an embedding of \pi_1(S) given by pushing the marked point around loops. We prove that for g>=3, the subgroup P(S) is the unique normal,…

First let $G$ be a completely solvable Lie group. We recall the proof of the following result: Any closed subgroup of $G$ possesses a unique syndetic hull in $G$. As a consequence we conclude that any uniform subgroup $\Gamma$ of $G$ is…

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically…

When $G$ is a connected compact Lie group, and $\pi$ is a closed surface group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset which is a smooth symplectic manifold. This symplectic structure is $Out(\pi)$-invariant…

For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing…

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic…