Related papers: Flexibility of surface groups in classical groups

We give an example of a Teichm\"uller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index,…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

If $\bar\rho$ is an automorphic modulo $p$ Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of $\bar\rho$. We prove new results in this direction in the case of a unitary group…

We observe that upper densities and spherical Federer densities may differ on all two dimensional surfaces of the sub-Riemannian Heisenberg group. This provides an entire class of intrinsic rectifiable sets having upper density strictly…

We prove that for N greater than or equal to 4, all smooth hypersurfaces of degree N in P^N are birationally superrigid. First discovered in the case N = 4 by Iskovskikh and Manin in a work that started this whole direction of research,…

Let $\Gamma$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p \geq 2$ and let $(X,\mu_X)$ be an ergodic standard Borel probability $\Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $\sigma: \Gamma \times X…

Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = {\l}\oplus \u$, where $\u$ is…

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the…

A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.

A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $\Gamma$ such that the quotient $G/\Gamma$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a…

We study the volume of maximal representations from a surface group into $\mathrm{SO}_0(2,3)$. We show that it is bounded from above, uniformly in the genus of the surface. We also prove that on the Gothen components, it is bounded from…

We consider a one-parametric series of left-invariant Lorentzian structures on the universal covering of the Lie group SL(2,R). These structures have SO(1,1)-symmetry and they are deformations of the anti-de Sitter Lorentzian manifold. We…

We generalize our methodology for computing with Zariski dense subgroups of $\mathrm{SL}(n, \mathbb{Z})$ and $\mathrm{Sp}(n, \mathbb{Z})$, to accommodate input dense subgroups $H$ of $\mathrm{SL}(n, \mathbb{Q})$ and $\mathrm{Sp}(n,…

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group, called a Beauville group. Here we consider which characteristically simple groups can be Beauville groups.…

In this paper, we utilize our previous results on mod p monodromy of cyclic coverings of the projective line to realize a large series of groups of the form PSL(n, q) and PSU(n, q) as Galois groups over Q. We achieve for the first time a…

A surface $\Sigma$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\Sigma) \to (M,\Sigma)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$…

We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which…

We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the…

For $n > 2$, let $\Gamma$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma$. This forms the main component of our…

This paper considers some algebraic surfaces that can deform to planar Zappatic stable surfaces with a unique singularity of type En. We prove that the Galois covers of these surfaces are all simply connected of general type, for n >= 4,…