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 $ϕ$ into a relatively hyperbolic group $G$ is relatively hyperbolic with respect to a system of subgroups whose image under $ϕ$ is situated in a uniformly bounded distance from the parabolic subgroups of $G$.
A continuous equivariant map from the Floyd boundary of a relatively hyperbolic group (RHG for short) to its Bowditch boundary is constructed. Such a map is unique unless the group is two-ended. In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions. In our approach some "visibility" conditions in graphs are essential. We introduce a class of "visibility actions" that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions. We …
In this paper, we study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that for the Floyd metric and shortcut metrics respectively, they are are both equal to a constant times the growth rate of the group. In the proof, we study a special class of conical points called uniformly conical points and establish that, in both boundaries, there exists a sequence of Alhfors regular sets with dimension tending to the Hausdorff dimension and these sets consist of uniformly conical points.
In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using dynamical methods. Some applications are discussed. The class of groups acting geometrically finitely on Floyd boundaries turns out to be easily understood. However, we also show that Dunwoody's inaccessible group does not act geometrically finitely on its Floyd boundary.
Let $p$ be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first $L^p$-cohomology space of some groups that have one end. We also make a connection between the first $L^p$-cohomology space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number $p$ such that the first $L^p$-cohomology space of a nonelementary hyperbolic group does not vanish.
We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nesting actions on $\R $-trees and we show that nesting actions with non overlapping translation intervals give rise to isometric actions.
Let $Γ$ be a relatively hyperbolic group and let $μ$ be an admissible symmetric finitely supported probability measure on $Γ$. We extend Floyd-Ancona type inequalities up to the spectral radius of $μ$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on $Γ$ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfie…
Given a sublinear function $κ$, $κ$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \fro…
It is well known that the Tits boundary of a proper cocompact CAT(0) space embeds into every asymptotic cone of the space. We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. The resulting projection from any asymptotic cone to the Tits boundary is determined by the visual topology; on the …
We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups with torsion. The main theorems are new in that they generalize results found in the literature, but a significant aim is expository. Toward that end, brief but reasonably comprehensive introductions to the theories of ANRs (absolute neighborhood retracts) and Z-sets are included, …
We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erdős--Rényi model. We show that at densities below $\left(\sqrt{\frac{1}{2}}-ε\right)\sqrt{\frac{\log{n}}{n}}$ random right-angled Coxeter groups almost surely have Morse hyperbolic surface subgroups. This implies their Morse boundaries contain embedded circles and they cannot be quasi-isometric to a right-angled Artin group. Further, at densities above $\left(\sqrt{\frac{1}{2}}+ε\right)\sqrt{\frac{\log{n}}{n}}$ we show that, almost surely, the hyperbolic Morse special subgroups of a random right-angled…
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a 2-stable, quasi-möbius homeomorphism between their Morse boundaries. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental group…
In this paper we study certain groups of bilipschitz maps of the boundary minus a point of a negatively curved space that is an abelian-by-cyclic solvable Lie group, where the extension is given by a matrix whose eigenvalues all lie outside of the unit circle. The case where the extension matrix is diagonal was previously studied by Dymarz. As an application, combined with work of Eskin-Fisher-Whyte and Peng, we provide the last steps in the proof of quasi-isometric rigidity for a class of lattices in solvable Lie groups.
We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing …
A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function $κ$, $κ$-Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that $κ$-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of…
We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric then the cores of their Tits boundaries are bi-Lipschitz.
In this paper we show that if $Y=N \times \mathbb{Q}_m$ is a metric space where $N$ is a Carnot group endowed with the Carnot-Caratheodory metric then any quasisymmetric map of $Y$ is actually bilipschitz. The key observation is that $Y$ is the parabolic visual boundary of a mixed type locally compact amenable hyperbolic group. The same results also hold for a larger class of nilpotent Lie groups $N$. As part of the proof we also obtain partial quasi-isometric rigidity results for mixed type locally compact amenable hyperbolic groups. Finally we prove a rigidity result for uniform subgroups of…
The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk. If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified wit…
In this paper we provide the final steps in the proof, announced by Eskin-Fisher-Whyte, of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable Lie groups.
We study discrete, cocompact, isometric actions of groups on Hadamard spaces, and the induced actions on ideal boundaries. For a class of groups generalizing fundamental groups of three-dimensional graph manifolds, we find a set of invariants for the action which determine the boundary action up to equivariant homeomorphism. This work was inspired by (and answers) a question of Gromov from "Asymptotic invariants of infinite groups."
The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application …
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric prop…
To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that …
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi--convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces. We use this approach, together with results of Leininger--Schleimer, to deduce that there is no purely geometr…
