Related papers: Distribution of lattice orbits on homogeneous vari…

In this survey, we gather together various results on the action of a real form of a complex semisimple Lie group on its flag manifolds. We start with the finiteness theorem of J.Wolf implying that at least one of the orbits is open. We…

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct…

For a cocompact group $\G$ of $\slr$ we fix a real non-zero harmonic 1-form $\alpha$. We study the asymptotics of the hyperbolic lattice-counting problem for $\G$ under restrictions imposed by the modular symbols $\modsym{\gamma}{\a}$. We…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

This paper is a survey article on the limiting behavior of the discrete spectrum of the right regular representation in $L^2(\Gamma\bs G)$ for a lattice $\Gamma$ in a reductive group $G$ over a number field. We discuss various aspects of…

We study the theory of convergence for CAT$(0)$-lattices (that is groups $\Gamma$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces) and their quotients (CAT$(0)$-orbispaces). We describe some splitting and collapsing…

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical…

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank…

We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of…

Let $\Gamma$ be a finitely generated cocompact lattice of a totally disconnected locally compact group $G$, and $C$ a dense subgroup of $G$ that contains and commensurates $\Gamma$. We study the problem of describing all finitely generated…

By using results by D. Witte on the superigidity of lattices in solvable Lie groups we get a different proof of a recent remarkable result obtained by D. Guan on the de Rham cohomology of a compact solvmanifold, i.e. of a quotient of a…

The $\gamma$-Cambrian semilattices $\mathcal{C}_{\gamma}$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, and they are lattices if and only if $W$ is…

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $\Gamma$ on a topological manifold where the…

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…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and \Gamma a lattice in G. We prove that every \Gamma-orbits in the Furstenberg boundary G/P is equidistributed…

Let $X$ be a locally symmetric space $\Gamma\backslash G/K$ where $G$ is a connected non-compact semisimple real Lie group with trivial centre, $K$ is a maximal compact subgroup of $G$, and $\Gamma\subset G$ is a torsion-free irreducible…

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index,…

Let $X$ be an orientable hyperbolic surface of genus $g\geq 2$ with a marked point $o$, and let $\Gamma$ be an orientable hyperbolic surface group isomorphic to $\pi_{1}(X,o)$. Consider the space $\text{Hom}(\Gamma,S_{n})$ which corresponds…

Answering a question by Chatterji--Dru\c{t}u--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space…