Related papers: Nonexistence of invariant rigid structures and inv…

We study rigidity properties of lattices in terms of invariant means and commensurating actions (or actions on CAT(0) cube complexes). We notably study Property FM for groups, namely that any action on a discrete set with an invariant mean…

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds…

Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an…

Let $\Gamma$ be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic, properand geodesic metric space. We study the action of the outer automorphism group Out($\Gamma$) onthe set X($\Gamma$,G) of conjugacy classes of…

A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of…

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in…

Local action principles on a manifold $\M$ are invariant (if at all) only under diffeomorphisms that preserve the boundary of $\M$. Suppose, however, that we wish to study only part of a system described by such a principle; namely, the…

For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study…

We outline an unified approach to geometrization of Lagrange mechanics, Finsler geometry and geometric methods of constructing exact solutions with generic off-diagonal terms and nonholonomic variables in gravity theories. Such geometries…

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space…

We study the dynamics of countable groups on their respective spaces of quasimorphisms. For cohomologically non-trivial quasimorphisms we show that there are no invariant measures and classify stationary measures. Within the equivalence…

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of L\"uck, dealing with towers of finitely sheeted normal coverings. We…

Let $\Gamma$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $\Gamma$ on $\R^d$ is strongly irreducible and that $\Gamma$ contains a proximal and expanding element. We describe contraction properties of the…

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On…

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $\Gamma<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $\Gamma$. This shows that almost all so-called "standard…

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in…

A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed,…

This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic…

There are a number of examples of variations of Hodge structure of maximum dimension. However, to our knowledge, those that are global on the level of the period domain are totally geodesic subspaces that arise from an orbit of a subgroup…