Related papers: Zimmer's conjecture for lattice actions: the ${\rm…

We establish a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions, when the action group is unimodular.

In \cite{Ghys} it is proved that any morphism from a subgroup of finite index of $\mathrm{SL}(n,\mathbb{Z})$ to the group of analytic diffeomorphisms of $\mathbb{S}^2$ has a finite image as soon as $n\geq 5$. The case $n=4$ is also claimed…

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…

We determine the existence of cocompact lattices in groups of the form $\V\rtimes\SL_2(\R)$, where $\V$ is a finite dimensional real representation of $\SL_2(\R)$. It turns out that the answer depends on the parity of $\dim(\V)$ when the…

Let $L$ be a lattice. We call a congruence relation $\gQ$ of $L$ isoform, if any two congruence classes of $\gQ$ are isomorphic (as lattices). Let us call the lattice $L$ isoform, if all congruences of $L$ are isoform. G. Gr\"atzer and…

For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study…

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that…

We study smooth actions by lattices in higher-rank simple Lie groups. Assuming one element of the action acts with positive topological entropy, we prove a number of new rigidity results. For lattices in $\mathrm{SL}(n,\mathbb{R})$ acting…

The Gr\"atzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable…

We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor iff A is an antilattice. We also generalize several other results of R. Kadison pertaining to infima and suprema in…

We study the action of a lattice in the group SL(2,R) on the plane. We obtain a formula which simultaneously describes visits of an orbit to either a fixed ball, or an expanding or contracting family of annuli. We also discuss the…

J. Tuma proved an interesting "congruence amalgamation" result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: --A.P. Huhn proved that every distributive algebraic lattice $D$ with…

A generalization of the Auslander conjecture is proved in the case when the Levi factor of the Zariski closure of the acting group is a product of simple real algebraic groups of rank \leq 1. Also, the Auslander conjecture is proved for…

The Guillemin-Sternberg conjecture states that "quantisation commutes with reduction" in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups $G$ acting on compact…

We study the behaviour of the minimal slope of Euclidean lattices under tensor product. A general conjecture predicts that $\mu_{min}(L \otimes M) = \mu_{min}(L)\mu_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that this is the…

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 give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on…

We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL…

We propose a lattice version of Chern-Simons gravity and show that the partition function coincides with Ponzano-Regge model and the action leads to the Chern-Simons gravity in the continuum limit. The action is explicitly constructed by…

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log…