Related papers: On geodesic ray bundles in hyperbolic groups

We show that if H is a quasiconvex subgroup of a hyperbolic group G then the relative Cayley graph Y (also known as the Schreier coset graph) for G/H is Gromov-hyperbolic. We also observe that in this situation if G is torsion-free and…

Let $\Gamma$ be a torsion-free subgroup of $SL_3(R)$ commensurable with $SL_3(Z)$, and $Y=SO_3(R)\backslash SL_3(R)/\Gamma$ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of…

The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the…

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a…

Given a closed surface $S$ with finitely generated Veech group $G$ and its $\pi_1(S)$-extension $\Gamma$, there exists a hyperbolic space $\hat{E}$ on which $\Gamma$ acts isometrically and cocompactly. The space $\hat{E}$ is obtained by…

Suppose that $\Sigma$ is a hyperbolic surface and $f:\mathbb R_+\to\mathbb R_+$ a monotonic function. We study the closure in the projective tangent bundle $PT\Sigma$ of the set of all geodesics $\gamma$ satisfying $I(\gamma,\gamma)\leq…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

It is shown that there are groups $\Gamma$ with finite generating sets $S$ such that the adjacency operator of the Cayley graph ${\rm Cay}(\Gamma,S)$ is a disjoint union of $N$ intervals, for arbitrarily large integers $N$.

Let $\mathbf{\Gamma} = (V,E)$ be a (non-trivial) finite graph with $\lambda: E \rightarrow \mathbb{R}_{+}$, an edge labelling of $\mathbf{\Gamma}$. Let $\rho : V\rightarrow \mathbb{R}^{2}$ be a map which preserves the edge labelling. The…

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…

In this paper, we construct a family of quasi-strongly regular Cayley graphs $\Gamma_H(G)$ which is defined on a finite group $G$ with respect to a subgroup $H$ of $G$. We also compute its full automorphism group and characterize various…

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

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…

For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic;…

This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on…

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $\Gamma$ is a finite graph which contains at least two vertices and is not a join and…

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The \textit{mixed Cayley graph} $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and…

We characterise hyperbolic groups in terms of quasigeodesics in the Cayley graph forming regular languages. We also obtain a quantitative characterisation of hyperbolicity of geodesic metric spaces by the non-existence of certain local…

The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an…

We construct cross sections for the geodesic flow on the orbifolds $\Gamma\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the…