Related papers: Relatively hyperbolic groups with free abelian sec…

For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an…

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\phi$ into a…

We give a topological interpretation of the free metabelian group, following the plan described in [Ver1,Ver2]. This interpretation is based on considering the Caley graph of a finitely generated group G as one-dimensional complex; its…

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is…

A group G has homological dimension less or equal to 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L^2-Betti numbers. We also show that a finitely generated elementary amenable group G…

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every…

This paper is devoted to the computation of the space $H_b^2(\Gamma,H;\mathbb{R})$, where $\Gamma$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in…

Our first result gives a partial converse to a well-known theorem of A. Ancona for hyperbolic groups. We prove that a group $G$, equipped with a symmetric probability measure whose finite support generates $G$, is hyperbolic if it is…

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of…

The Groups of causal and conformal automorphisms of globally hyperbolic spacetimes were studied. In two dimensions, we prove that all globally hyperbolic spacetimes that are directed and connected are causally isomorphic. We work out the…

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic…

Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely…

Let $G$ be a dp-minimal group; we prove some consequences of several different hypotheses on $G$. First, if $G$ is torsion-free, then it is abelian. Second, if $G$ admits a distal f-generic type, then it is virtually nilpotent; we prove…

We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative…

Given isometric actions by a group G on finitely many \delta-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a…

In this article, we prove a version of Martin and Skora's conjecture that convergence groups on the $2$-sphere are covered by Kleinian groups. Given a relatively hyperbolic group pair $(G,\mathcal{P})$ with planar boundary and no Sierpinski…

We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of…

A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie…

We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex,…

The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each…