arXiv research

Proof that specific groups are quasi-isometrically rigid.

Study shows certain infinite-ended groups are not quasi-isometrically rigid.

New groups are hyperbolic and rigid in mapping class groups.

We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.

Research shows arc complex is not quasi-isometric to sphere complex.

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…

New rigidity theorem for product of lattices.

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 always quasi-isometric, and a group G is quasi-isometrically rigid if every group …

Let $Γ$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $Γ$ is abstractly commensurable with $Γ$.

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 punctured surface S of genus at least 2: if K is a finitely generated group quasi-i…

In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the h…

The paper explores conditions for quasi-isometric subgroups and their filtered ends.

Let S be an oriented surface of finite type of genus g with m punctures and where 3g-3+m>1. We show that the mapping class group M(S) of S is quasi-isometrically rigid. We also give a different proof of the following result of Behrstock and Minsky: The homological dimension of the asmyptotic cone of M(S) of S equals 3g…

We prove that the every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.

Random quotients of mapping class groups have rigid properties.

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of…

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

The study shows how to embed cusp-decomposable manifolds quasi-isometrically.

Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions ar…

We prove that PSL(2,Z[1/p]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and, independent of p, the quasi-isometry group of PSL(2,Z[1/p]) is PSL(2,Q). In addition, we char…

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ wh…

The study examines the rigidity of mapping class groups under large powers of twists.

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group.…

Study automorphism groups of Artin groups, proving rigidity and classification results.

New examples of embeddings defy Anosov representation limits.

In this paper, we introduce a particularly nice family of locally CAT(-1) spaces, which we call hyperbolic P-manifolds. For $X^3$ a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of $X^3$ are characterized topologically. Straightforwar…

The paper studies a group action on a hyperbolic space derived from a lattice Veech group.

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admi…

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 negatively curved manifolds. A main conjecture provides a general description; an extend…

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twi…

Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian mani…

Survey on quasi-isometries of group pairs and their invariants.

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 left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namel…

Study shows saddle connection graph's geometry and quasi-isometry properties.

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…

An elementary proof shows that quasi-isometric groups to integers are virtually integers.

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflect…

We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflat…

Leighton's graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton's theorem that allows generalizations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups w…

Let $G$ be a right-angled Artin group with defining graph $Γ$ and let $H$ be a finitely generated group quasi-isometric to $G(Γ)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $Γ$ does not have induced 4-cycle; (3) $Γ$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is…

Quasi-isometries in horospherical products are close to product maps.

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove that if $M$ is i…

Graphically discrete groups have strong rigidity properties.

Let $S$ be a projective plane with $3$ holes. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets. As a corollary, we obtain that the group of simplicial automorphisms of $\mathcal{C}(S)$ is isomorphic to the mapping class group $\mathrm{Mod}(S)$. We also prove …

We study the large scale geometry of the upper triangular subgroup of PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a lar…

Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $δ$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend…