Hyperbolicity Research Articles

Teichmüller displacement theorem on Gromov hyperbolic spaces

Abstract Given a Gromov hyperbolic domain $G\subsetneq \mathbb{R}^n$ with uniformly perfect Gromov boundary, Zhou and Rasila recently proved that for all quasiconformal homeomorphisms $\psi\colon G\to G$ with identity value on the Gromov boundary, the quasihyperbolic displacement $k_G(x,\psi(x))$ for all $x\in G$ is bounded above. In this paper, we generalize this result and establish Teichmüller displacement theorem for quasi-isometries of Gromov hyperbolic spaces in a quantitative way. As applications, we obtain its connections to bilipschitz extensions of certain Gromov hyperbolic spaces.

Uniform Metric Graphs

We prove that every complete metric space “is” the boundary of a uniform length space whose quasihyperbolization is a geodesic visual Gromov hyperbolic space. There is a natural quasimöbius identification of the original space’s conformal gauge with the canonical gauge on the Gromov boundary. All parameters are absolute constants.

Barycenters and a law of large numbers in Gromov hyperbolic spaces

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.

Relative L^p-cohomology and application to Heintze groups

We introduce the notion of relative \(L^p\)-cohomology as a quasi-isometry invariant defined for a Gromov-hyperbolic space and a point on its boundary at infinity and reproduce some basic properties of \(L^p\)-cohomology in this context. In the case of degree 1 we show a relation between the relative and the classical \(L^p\)-cohomology. As an application, we explicitly construct non-zero relative \(L^p\)-cohomology classes for a purely real Heintze group of the form \(\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}\), which gives a way to prove that the eigenvalues of \(\alpha\), up to a scalar multiple, are invariant under quasi-isometries.

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

We study the interplay between the backward dynamics of a non-expanding self-map f of a proper geodesic Gromov hyperbolic metric space X and the boundary regular fixed points of f in the Gromov boundary as defined in [8]. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behavior of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains Ω⊂⊂Cq, where Ω is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with q=2, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc D⊂C obtained by Bracci and Poggi-Corradini in [14,27,28]. In particular, with our geometric approach we are able to answer a question, open even for the unit ball Bq⊂Cq (see [5,26]), namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.

LINEAR GROWTH OF TRANSLATION LENGTHS OF RANDOM ISOMETRIES ON GROMOV HYPERBOLIC SPACES AND TEICHMÜLLER SPACES

AbstractWe investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

Random walks on mapping class groups

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks and vice versa. Our emphasis is on the analogues of classical theorems such as laws of large numbers and central limit theorems and the properties of harmonic measures under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.

Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

We study random walks on the isometry group of a Gromov hyperbolic space or Teichmüller space. We prove that the translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment. While doing this, we recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. Also discussed are the corresponding laws of the iterated logarithm. Finally, we prove sublinear geodesic tracking by random walks with finite (1/2)-th moment and logarithmic tracking by random walks with finite exponential moment.

Equivalent topologies on the contracting boundary

The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the \(\kappa\)–Morse boundary with a sublinear function \(\kappa\). The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the \(\kappa\)–Morse boundary contains the contracting boundary. We will prove this conjecture: when \(\kappa =1\) is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.

Random walks on hyperbolic spaces: second order expansion of the rate function at the drift

Let (X,d) be a separable geodesic Gromov-hyperbolic space, o∈X a basepoint and μ a countably supported non-elementary probability measure on Isom(X). Denote by z n the random walk on X driven by the probability measure μ. Supposing that μ has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence 1 nd(z n ,o) and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence d(z n ,o). This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of d(z n ,o) at the origin using a martingale decomposition first introduced by Benoist–Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.

The horofunction boundary of a Gromov hyperbolic space

We highlight a condition, the approaching geodesics property, on a proper geodesic Gromov hyperbolic metric space, which implies that the horofunction compactification is topologically equivalent to the Gromov compactification. It is known that this equivalence does not hold in general. We prove using rescaling techniques that the approaching geodesics property is satisfied by bounded strongly pseudoconvex domains of $$\mathbb {C}^q$$ endowed with the Kobayashi metric. We also show that bounded convex domains of $$\mathbb {C}^q$$ with boundary of finite type in the sense of D’Angelo satisfy a weaker property, which still implies the equivalence of the two said compactifications. As a consequence we prove that on those domains big and small horospheres as defined by Abate in (Math Z 198(2):225–238, 1988) coincide. Finally we generalize the classical Julia’s lemma, giving applications to the dynamics of non-expanding maps.

Uniform Poincaré inequalities on measured metric spaces

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincar\'e inequalities on $(X,d,\mu)$ if it satisfies a local Poincar\'e inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a $(\sigma,\beta,\sigma)$-Poincar\'e inequality. If $(X,d,\mu)$ is a $\delta$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincar\'e inequality with exponential growth of the Poincar\'e constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space then it supports a uniform Poincar\'e inequality and the Poincar\'e constant depends on the growth of the fundamental group.

Geometric characterizations of Gromov hyperbolic Hölder domains

Abstract In this paper, we investigate Hölder continuity of quasiconformal mappings in ℝ n {\mathbb{R}^{n}} from the points of view of quasihyperbolic geometry and the theory of Gromov hyperbolic spaces. We establish several characterizations of Gromov hyperbolic domains satisfying the Gehring–Martio-type quasihyperbolic boundary conditions. As applications, we generalize certain results concerning Hölder continuity of conformal mappings, establishing counterparts of results of Becker and Pommerenke, Smith and Stegenga, and Näkki and Palka in higher-dimensional Euclidean spaces.

Sublinearly Morse boundary I: CAT(0) spaces

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.We attempt to construct an analogue of the Gromov boundary that encodes the hyperbolic directions in a metric space. To this end, for any sublinear function κ, we define a subset of the visual boundary called the κ–Morse boundary. We show that, equipped with a coarse notion of visual topology, this space is QI-invariant and metrizable. That is to say, the κ–Morse boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it is shown in the Appendix that the Poisson boundary of random walks is naturally identified with the (tlog⁡t)–boundary.

Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Lǐsic case* *The authors were partially supported by CONICYT PIA ACT172001 and by Proyecto FONDECYT 1180922.

We consider the reducibility problem of cocycles by isometries of Gromov hyperbolic metric spaces in the Lǐsic setting. We show that provided that the boundary cocycle (that acts on a compact space) is reducible in a suitable Hölder class, then the original cocycle by isometries (that acts on an unbounded space) is also reducible.

Potential Theory on Gromov Hyperbolic Spaces

Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.

Sphericalizations and applications in Gromov hyperbolic spaces

In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that doubling properties regarding two classes of metrics on the Gromov boundaries of hyperbolic spaces are equivalent. Second, we obtain a characterization of unbounded Gromov hyperbolic domains via the metric space sphericalization. Finally, we show that there is a homeomorphic correspondence between the Gromov boundary and the inner metric boundary of a Gromov hyperbolic domain which is φ-uniform.

Second bounded cohomology and WWPD

Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a 'really' weak property discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.

Generalized hyperbolicity and shadowing in L spaces

It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known as generalized hyperbolic operators. This class of operators seems to be an important bridge between hyperbolicity and the shadowing property. In this article, we show that for a large natural class of operators on L p ( X ) the notion of generalized hyperbolicity and the shadowing property coincide. We do this by giving sufficient and necessary conditions for a certain class of operators to have the shadowing property. We also introduce computational tools which allow construction of operators with and without the shadowing property. Utilizing these tools, we show how some natural probability distributions, such as the Laplace distribution and the Cauchy distribution, lead to operators with and without the shadowing property on L p ( X ) .