Related papers: Dehn functions and the space of marked groups

An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups. However, not much is…

We investigate oriented graphs based on the curve complex $C(S)$ of a closed surface $S$ and induced by functions on the vertex set of $C(S)$. In particular, we introduce the Dehn quasi-homothetic functions, which behave similarly to…

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. The main theorem of the paper is as follows. Let G be a finitely generated, large group and let g_1,...,g_r be a…

Let $G$ be the multiplicative group generated by the gamma functions $\Gamma(ax+1)$ $(a=1,2,\dots)$, and $H$ be the subgroup of all elements of $G$ that converge to nonzero constants as $x\rightarrow\infty$. The quotient group $G/H$ is the…

The Kauffman bracket skein module $K(M)$ of a $3$-manifold $M$ is the quotient of the $\mathbb{Q}(A)$-vector space spanned by isotopy classes of links in $M$ by the Kauffman relations. A conjecture of Witten states that if $M$ is closed…

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there…

This paper studies the locally uniform exponential growth and product set growth for a finitely generated group $G$ acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing…

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We…

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups…

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed…

If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a…

Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let $v$ be a vertex of $T$. Let $({X_v},d_v)$ denote the hyperbolic metric space corresponding to $v$. Then $i : X_v \rightarrow X$…

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…

Using the $\mathbb{R}((X))$-measure, we define and study certain $\mathbb{C}((X))$-valued functions on $\mathrm{GL}_n(F)$ for $F$ a two-dimensional local field. In particular, we define a convolution product on such suitable functions,…

We show, using the Kobayashi and Caratheodory metrics on special holomorphic disks in the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe…

In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly…

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of…

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $\delta$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative…

We introduce the notion of finite stature of a family $\{H_i\}$ of subgroups of a group $G$. We investigate the separability of subgroups of a group $G$ that splits as a graph of hyperbolic special groups with quasiconvex edge groups. We…

In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery…