Related papers: Thin actions on CAT(0) spaces

Let $\Gamma$ be a discrete countable group acting isometrically on a measurable field $\mathbf{X}$ of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability $\Gamma$-space $(\Omega,\mu)$. If $\mathbf{X}$…

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be…

We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic…

We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors…

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the…

We introduce two new tools for studying CAT(0) spaces: \emph{curtains}, versions of cubical hyperplanes; and the \emph{curtain model}, a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a…

We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and…

We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such…

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and…

In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at…

If G is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces E_{vc} and E_{fbc} under the additional assumption that the action of G has a well-behaved collection of…

It is well-known that $\mathrm{SL}_{n}(\mathbf{Q}_{p})$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{CAT}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{CAT}(0)$ spaces on which…

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times…

If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the…

We show that a linear group without unipotent elements of infinite order possesses properties akin to those held by groups of non positive curvature. Moreover in positive characteristic any finitely generated linear group acts properly and…

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain…

The notions of nonpositive curved spaces and biautomatic groups are generalizations of the geometric properties of hyperbolic spaces and computational properties of their fundamental groups. Given the mutual origins of these conditions, one…

Let $G$ be the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $\Gamma \subset G$ on the boundary of $X$, which is…

We prove that an IRS of a group with a geometrically dense action on a CAT(0) space also acts geometrically densely; assuming the space is either of finite telescopic dimension or locally compact with finite dimensional Tits boundary. This…

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…