Related papers: Fundamental groups of asymptotic cones

We generalise results about isometric group actions on metric spaces and their fundamental regions to the context of merely continuous group actions. In particular, we obtain results that yield the relative compactness of a fundamental…

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Non-trivial examples appear in the context of Lie…

We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of…

Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup and a convenient arithmetic subgroup $\Gamma\subset G({\Bbb Q})$, one constructs an arithmetic manifold $S=S(\Gamma)=\Gamma\backslash X$. If $H\subset…

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the…

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not…

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_\lambda ,\lambda \in \Lambda \} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_\lambda ,\lambda \in…

The group $Ham(M,\omega)$ of all Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry…

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $\pi_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of…

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact…

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental…

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.

We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into…

In this paper, we show that the strong embeddability has fibering permanence property and is preserved under the direct limit for the metric space. Moreover, we show the following result: let $G$ is a finitely generated group with a coarse…

General arguments of Baumslag and Bieri guarantee that any metabelian group of finite Pr\"ufer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the…

We construct a space $\mathbb{P}$ for which the canonical homomorphism $\pi_1(\mathbb{P},p) \rightarrow \check{\pi}_1(\mathbb{P},p)$ from the fundamental group to the first \v{C}ech homotopy group is not injective, although it has all of…

It is proven that if $G$ is a finite group, then $G^\omega$ has $2^{\mathfrak c}$ dense nonmeasurable subgroups. Also, other examples of compact groups with dense nonmeasurable subgroups are presented.

The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation…

I show that one can explicitly construct topologically/geometrically distinguishable data which provide isomorphic copies (i.e. \emph{isomorphs}) of the tempered fundamental group of a geometrically connected, smooth, quasi-projective…