Related papers: Martin boundary covers Floyd boundary

For non-compact, locally symmetric moduli spaces M, the set of geodesics and the geometry of the boundary can be completely characterised using group theory. In particular, geodesics that asymptote to a given infinite distance boundary…

We develop the theory of Patterson-Sullivan measures on the boundary of a locally compact hyperbolic group, associating to certain left invariant metrics on the group measures on the boundary. We later prove that for second countable,…

We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $\Gamma$ admits a finite…

We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also…

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively…

We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Gamma. We consider all walks of length N on G, starting from v_i and ending at v_j To each such walk $w$…

We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Bj\"orklund and the second-named author. Our theory is based on the notion of a quasi-isometric quasi-action (qiqac) of an…

We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm…

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries \Gamma whose limit set is uniformly perfect, and a disjoint collection of horoballs {H_j}, we show that the set of limit points badly approximable…

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $\mu$ with a…

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a…

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined…

This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup…

For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such…

We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary.…

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial''…

We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume…

On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with…

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result…

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log,2}(\partial\Gamma)\subset L^2(\partial\Gamma)$, the domain of the logarithmic Laplacian on $\partial\Gamma$. We show that…