Related papers: Mapping class group actions on configuration space…

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two…

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),\Omega Y]\cong \lim\limits_{\leftarrow} [J_n(X),\Omega Y]$ for any space $Y$. When…

Let M denote the mapping class group of S, a compact connected oriented surface with one boundary component. The action of M on the nilpotent quotients of the fundamental group of S allows to define the so-called Johnson filtration and the…

Let $S$ be a surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let $\mathcal{MF}$ be the space of equivalence classes of measured foliations of…

Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it fixes a point.

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $\phi:{\rm Mod}_g\to {\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has…

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 $g+n>3$.

We show that the first Johnson subgroup of the mapping class group of a surface S of genus greater than one acts ergodically on the moduli space of representations of the fundamental group of S in SU_2. Our proof relies on a local…

We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We conclude that…

We prove the following well known conjecture: let $\Sigma$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $\phi \in \textup{Mod}(\Sigma)$ be a an infinite order mapping class. Then there exists…

Let $\Sigma$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $\Sigma$ emanating from the puncture. Then there is a natural action of the mapping class…

Let $\Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of…

We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let \ $G=E_{n}(R)$, $EU_{n}(R,\Lambda ),$ $\mathrm{SAut}(F_{n})$ or $\mathrm{SOut}(F_{n}).$ As applications, we prove that when…

The algebras $\mathcal{L}_{g,n}(H)$ have been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the middle of the 1990's, in the program of combinatorial quantization of the moduli space of flat connections over the surface…

Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its…

Let $\mathcal{M}_{0,n+1}$ be the moduli space of genus zero Riemann surfaces with $n+1$ marked points. In this paper we compute $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p)$ and $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p(\pm 1))$…

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of…

We describe the mapping class group action on the cohomology of the twisted $\mathrm{SL}_n$-character variety of a surface $\Sigma_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this…

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the…

The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup $K_{g,1}$ of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of…