Related papers: Mapping class groups of surfaces of genus $\geq 3$…

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus,…

We provide new constraints for algebro-geometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction…

We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface…

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…

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map…

We prove that the mapping class group of a closed oriented surface of genus $\rho \ge 3$ has no proper subgroup of index $\le 4 \rho +4$.

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous…

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…

In this note, we prove that the compactly supported mapping class group of a surface containing a genus $3$ subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order $6$…

In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit…

Let X be a K3 surface with a polarization H with H^2=2rs. Assume that H.N(X)=Z for the Picard lattice N(X). The moduli space Y of sheaves over X with the Mukai vector (r,H,s) is again a K3 surface. We prove that Y\cong X, if there exists…

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many…

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a…

Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says…

Let $\Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $\Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that…

We classify homomorphisms from mapping class groups by using finite subgroups. First, we give a new proof of a result of Aramayona--Souto that homomorphisms between mapping class groups of closed surfaces are trivial for a range of genera.…

Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$. Let $\Gamma(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of…

For $\Sigma$ an orientable surface of finite topological type having genus at least 3 (possibly closed or possibly with any number of punctures or boundary components), we show that the mapping class group $Mod(\Sigma)$ has no faithful…

It has been known since the time of Nielsen that the mapping class group $\text{Mod}_{g,1}$ of a surface of genus $g$ and one puncture acts faithfully by homeomorphisms on the circle. In this note, we show that this standard representation…