Related papers: The mapping class group cannot be realized by home…

We completely describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

In all possible cases, we prove that local embeddings between two curve complexes whose complexities do not increase from domain to codomain are induced by surface homeomorphism. This is our first main result. From this we can deduce our…

Suppose that $M$ is a connected orientable $n$-dimensional manifold and $m>2n$. If $H^i(M,\R)=0$ for $i>0$, it is proved that for each $m$ there is a monomorphism $H^m(W_n,\on{O}(n))\to H^m_{\on{cont}}(\on{Diff}M,\R)$. If $M$ is closed and…

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular…

Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$\rho: \pi_1(\Sigma_{g,n})\to…

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from…

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in…

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…

We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the…

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification…

Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class…

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms of it to the mapping class group of it has no…

We prove the congruence subgroup property for the centralizer of a finite subgroup $G$ in the mapping class group of a hyperbolic oriented and connected surface of finite topological type $S$ such that the genus of the quotient surface…

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$,…

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product…

A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the…

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is…

Let $N_g$ be a closed, connected, nonorientable surface of genus $g$. We prove that for $g \ge 13$, the mapping class group $\text{Mod}(N_g)$ can be generated by exactly two elements. This improves the previously known bound of $g \ge 19$.

One of the main problems of the theory of dynamical systems is the determination of the existence of periodic orbits of a self-map and more generally, the structure of the set of periods. Define the minimum period of a class os self-maps of…

There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows…