Related papers: A rigidity result for normalized subfactors

Let $\Gamma$ be a weakly irreducible higher rank lattice. In this paper, we will prove various rigidity results for the $\Gamma$-action following a philosophy of the Zimmer program. We provide new rigidity results including local and global…

We show that for a typical high rank arithmetic lattice $\Gamma$, there exist finite index subgroups $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{1} \not\simeq \Gamma_{2}$ while $\widehat{\Gamma_{1}} \simeq \widehat{\Gamma_{2}}$. But…

If $G$ is a semisimple Lie group of real rank at least 2 and $\Gamma$ is an irreducible lattice in $G$, then every homomorphism from $\Gamma$ to the outer automorphism group of a finitely generated free group has finite image.

If $\Gamma$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example, $SL_n(\mathbb{Z})$, $n \geq 3$) and $\Lambda$ is a finitely generated group that is elementarily equivalent to $\Gamma$, then…

Let G be any locally compact, unimodular, metrizable group. The main result of this paper, roughly stated, is that if F<G is any finitely generated free group and \Gamma < G any lattice, then up to a small perturbation and passing to a…

Let $\Gamma$ be a group acting on with finite stabilizers and finite fundamental domain on a building of type $\tilde A_2$. We prove that any non-trivial normal subgroup of $\Gamma$ is of finite index in $\Gamma$.

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…

A locally compact group $G$ is a cocompact envelope of a group $\Gamma$ if $G$ contains a copy of $\Gamma$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $\Gamma,\Lambda$ having a common…

The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many…

We introduce the notion of L^2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that…

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $\Gamma$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely,…

Let $G$ be a real centre-free semisimple Lie group without compact factors. I prove that irreducible lattices in $G$ are rigid under two types of sublinear distortions. The first result is that the class of lattices in groups that do not…

We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $\Gamma$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $\Gamma$…

Let $\Gamma$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $\Gamma$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $\Gamma$ is…

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$-torsion element. We define…

Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete…

Given an irreducible lattice $\Gamma$ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma$-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal{L}(\Gamma)$, and for the…

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…

We prove that if $\Gamma$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(\Gamma)$ is prime. In particular, we deduce that the II$_1$ factors…

Let $\Gamma$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $\Gamma$ on a $\CAT(0)$ cubical complex has a global fixed point.