Related papers: Spectral gap for products and a strong normal subg…

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $\Gamma < G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic…

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups…

We show that if $G$ is a real semi-simple Lie group, and $\Gamma$ is a discrete subgroup of $G$ containing a subgroup $\Sigma$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $\Gamma$ is not isomorphic to a…

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was…

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation $R$-lattice for the finite $p$-group $G$ in terms of the restriction to a normal subgroup $N$ and…

We prove that for any finite index subgroup $\Ga$ in $SL_n(\mathbb{Z})$, there exists $k=k(n)\in\mathbb{N}$, $\ep=\ep(\Ga)>0$, and an infinite family of finite index subgroups in $\Ga$ with a Kazhdan constant greater than $\ep$ with respect…

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 $\Gamma$.

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the…

We show that a topologically generating set $S$ of a connected compact Lie group $G$ of size larger than a fixed polynomial in the rank of $G$ must be redundant (i.e., some proper subset of $S$ still topologically generates $G$). Similar…

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible…

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\Gamma\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\Gamma$ is virtually a product of…

We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of…

Let $S_n$ denote the symmetric group on $n$ elements, and $\Sigma\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $\Sigma$ is a…

Let $G$ be a locally compact group and $\mu$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(\pi, \cal H)$ of $G,$ we study spectral properties of…

We show that stationary characters on irreducible lattices $\Gamma < G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation…

A discrete group which admits a faithful, finite dimensional, linear representation over a field $\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky…

The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain states that the group $\mathrm{EL}_n(\mathcal{R})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{R}$, has Kazhdan's property…

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $\Lambda$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every…

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the…

In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly…