Related papers: Limit points of one-parameter subgroups for additi…

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property…

We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative…

Let $X$ be a projective hypersurface that is not a cone. The symmetrizer group of $X$ is an algebraic group parametrizing hypersurfaces whose Jacobian ideal coincides with that of $X$. We show that if the locus of points in $X$ with…

We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the…

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X under the action of G is a regular embedding, a condition satisfied in particular by smooth toric varieties and…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…

According to the work of Laitinen, Morimoto, Oliver and Pawa\l{}owski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to…

For a finite group $G$ not of prime power order, Oliver (1996) has answered the question which manifolds occur as the fixed point sets of smooth actions of $G$ on disks (resp., Euclidean spaces). We extend Oliver's result to compact (resp.,…

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by…

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a…

A topological group $G$ is B-amenable if and only if every continuous affine action of $G$ on a bounded convex subset of a locally convex space has an approximate fixed point. Similar results hold more generally for slightly uniformly…

We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remove the usual assumptions of properness and smoothness of the…

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety,…

By a commutative action on a smooth quadric $Q_n$ in $P^{n+1}$ we mean an effective action of a commutative connected algebraic group on $Q_n$ with an open orbit. We show that for $n \geq 3$ all commutative actions on $Q_n$ are additive…

We study a class of group actions on hyperk\"ahler manifolds which we call actions of linear type. If $M$ is a hyperk\"ahler manifold possessing such a $G$-action, the hyperk\"ahler Kirwan map is surjective if and only if the natural…

In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the…

For every countable abelian group $G$ we find the set of all its subgroups $H$ ($H\leq G$) such that a typical measure-preserving $H$-action on a standard atomless probability space $(X,\mathcal{F}, \mu)$ can be extended to a free…

We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a…

We show that any isometric action of a residually finite group admits approximate local finite models. As a consequence, if $G$ is residually finite, every isometric $G$-action embeds isometrically into a metric ultraproduct of finite…

Smith theory says that the fixed point of a semi-free action of a group $G$ on a contractible space is ${\bb Z}_p$-acyclic for any prime factor $p$ of $G$. Jones proved the converse of Smith theory for the case $G$ is a cyclic group acting…