Related papers: Approximate marked length spectrum rigidity in coa…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…

We prove measure rigidity for the action of (maximal) horospherical subgroups on homogeneous spaces obtained by quotient by a uniform (nonuniform) arithmetic lattices over a field of positive characteristic.

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any…

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions. We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that,…

For a smooth expanding map $f$ of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of $f$. This spectrum is analogous to the set of lengths of all closed geodesics on…

Kernel matrices are of central importance to many applied fields. In this manuscript, we focus on spectral properties of kernel matrices in the so-called ``flat limit'', which occurs when points are close together relative to the scale of…

We introduce asymptotically rigid mapping class groups of handlebodies and determine their finiteness properties, which vary depending on the space of ends of the underlying handlebody. As it turns out, in some cases, the homology of these…

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric.…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial…

Let $\Sigma$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated "framed mapping class group" on the homology of $\Sigma$ relative to its boundary…

We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.

The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we…

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in…

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a…

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is…

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a…

We consider the Gaussian ensembles of random matrices and describe the normal modes of the eigenvalue spectrum, i.e., the correlated fluctuations of eigenvalues about their most probable values. The associated normal mode spectrum is…

We show a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves.…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…