The goal of this paper is two-fold. First, we introduce and analyze a new relationship
between (Zariski-dense) abstract subgroups of the groups of F-rational points of two …

We show that if M is an arithmetic hyperbolic 3-manifold, the set QL (M) of all rational
multiples of lengths of closed geodesics of M both determines and is determined by the …

The article contains a survey of our results on length-commensurable and isospectral locally
symmetric spaces and of related problems in the theory of semisimple algebraic groups. We …

In this paper we show that totally geodesic subspaces determine the commensurability class
of a standard arithmetic hyperbolic $ n $-orbifold, $ n\ge 4$. Many of the results are more …

The purpose of this article is to produce effective versions of some rigidity results in algebra
and geometry. On the geometric side, we focus on the spectrum of primitive geodesic …

Let $ G $ be a topological group acting on a simplicial complex $\mathcal {X} $ satisfying
some mild assumptions. For example, consider a $ k $-regular tree and its automorphism …

ISOSPECTRAL CAYLEY GRAPHS OF SOME FINITE SIMPLE GROUPS Page 1 ISOSPECTRAL
CAYLEY GRAPHS OF SOME FINITE SIMPLE GROUPS ALEXANDER LUBOTZKY, BETH …

In this paper, we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic
3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally …

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must
be commensurable. Building toward a negative answer to this question, we construct …

In this article, we construct closed, isospectral, non-isometric locally symmetric manifolds.
We have three main results. First, we construct arbitrarily large sets of closed, isospectral …