Related papers: Quasi-isometric rigidity for the solvable Baumslag…

We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any…

We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly…

We show that the Lipschitz free space of a countable simplicial quasi-tree is isomorphic to $\ell^1$. As a consequence, every finitely generated group with Property (QT) of Bestvina--Bromberg--Fujiwara has a proper uniformly Lipschitz…

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…

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be…

We study quasiconformal maps on non-rigid Carnot groups equipped with Carnot metric. We show that for most non-rigid Carnot groups N, all quasiconformal maps on N must be biLipschitz.

We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces \cite{solecki1} and obtain the following exact equivalence: any action of a discrete group $\Gamma$ by isometries of a metric space…

We look at isometric actions on arbitrary hyperbolic spaces of generalised Baumslag - Solitar groups of arbitrary dimension (the rank of the free abelian vertex and edge subgroups). It is known that being a hierarchically hyperbolic group…

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $\Gamma$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely,…

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first…

We show that the geometric and homological finiteness properties of group pairs are invariant under a suitable notion of quasi-isometry for group pairs.

Let $BS(1,n)= <a,b : a b a ^{-1} = b ^n>$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces with (pseudo-)Anosov elements. That is, we consider a closed…

We introduce the class of extended admissible groups, which include both fundamental groups of non-geometric 3-manifolds and Croke-Kleiner admissible groups. We show that the class of extended admissible groups is quasi-isometrically rigid.

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are…

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is…

We prove that the rational number |n/m| is an invariant of the group von Neumann algebra of the Baumslag-Solitar group BS(n,m). More precisely, if L(BS(n,m)) is isomorphic with L(\BS(n',m')), then |n'/m'| = |n/m| or |m/n|. We obtain this…

We prove that a uniquely 2-divisible group that admits an almost regular involutory automorphism is solvable.

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost…