Related papers: Approximate marked length spectrum rigidity in coa…

We provide a combinatorial characterisation of positive diagrams satisfying the equality in the Morton-Franks-Williams bound for the degrees of the HOMFLY-PT polynomial. This characterisation allows generating with relative ease examples of…

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to…

It is proved that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. We provide two distinct proofs, one tailored to mapping class groups, and one applying to a larger…

The objective of this series is to study metric geometric properties of disjoint unions of amenable Cayley graphs by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint…

We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For…

We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid…

Let $g, n \geq 0$ and $\Sigma = \Sigma_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $\Sigma$ has a rigid relative automorphism group, whose elements fix each monodromies…

We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and…

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic…

In this paper we study the minimal and maximal $L^{2}$-cohomology of oriented, possibly not complete, Riemannian manifolds. Our focus will be on both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we will prove that…

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where…

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with…

We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of…

Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by…

Using methods originating in the theory of intersection spaces, specifically a de Rham type description of the real cohomology of these spaces by a complex of global differential forms, we show that the Leray-Serre spectral sequence with…

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous…

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the…

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been…