Related papers: Arithmetic hyperbolicity: automorphisms and persis…

We show that generic automorphisms of stable groups are supertight in a strong sense. In particular, we obtain the existence of supertight automorphisms. We also answer a question concerning the relationship between supertight automorphisms…

Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…

We consider rational projective homogeneous varieties over an algebraically closed field of positive characteristic, namely quotients of a semi-simple group by a possibly non-reduced parabolic subgroup. We determine the group scheme…

We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with…

By constructing, in the relative case, objects analoguous to Rips and Sela's canonical representatives, we prove that the set of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic…

We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism…

We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic…

Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…

Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear…

In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over…

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian…

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be…

Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be…

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension $K$ of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a…

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Using methods of p-adic analysis we give a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. More precisely, we show that any finitely generated torsion subgroup…

We prove a conjecture of Heath-Brown on the number of rational points of bounded height for a large class of projective varieties.

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve,…

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold…