Related papers: Approximate marked length spectrum rigidity in coa…

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge3$ with (finite or countably many) conical singularities $\{z_i\}_{i\in\mathfrak I}$ in the neighborhood of which the largest lower…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly…

Let ${\mathcal M}_g$ be the moduli space of compact connected Riemann surfaces of genus $g\geq 2$ and let $\widehat{{\mathcal M}_g}$ be its Deligne-Mumford compactification, which is stratified by the topological type of the stable Riemann…

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the…

Two planar embedded circle patterns with the same combinatorics and the same intersection angles can be considered to define a discrete conformal map. We show that two locally finite circle patterns covering the unit disc are related by a…

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…

The aim of the present paper is to study the distributions of the length multiplicities for negatively curved locally symmetric Riemannian manifolds. In Theorem 2.1, we give upper bounds of the length multiplicities and the square sums of…

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric…

A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that admit some non-trivial symmetry. When $d=2$ there is a large literature on this…

We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover,…

In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once…

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of…

Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees…

A subspace or subgroup is isotropic under a bilinear map if the restriction of the map on it is trivial. We study maximal isotropic subspaces or subgroups under skew-symmetric maps, and in particular the isotropy index---the maximum…

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of…

In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold…

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit…