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We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective…
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 study parabolic semigroups of finite shift in the unit disk with regard to the rate of convergence of their orbits to the Denjoy--Wolff point. We examine this rate in terms of Euclidean distance, hyperbolic distance and harmonic measure.…
In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for…
In this paper, we study analytic self-maps of the unit disk which distort hyperbolic area of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, M\"obius…
We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for $n\geq 2$, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then $f$ is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to…
We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic…
The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.
We consider a coarse version of the marked length spectrum rigidity: given a group with two left invariant metrics, if the marked length spectrum (the translation length function) under the two metrics are the same, then the two metrics are…
We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for…
We give improved bounds for the distortion of the Hausdorff dimension under quasisymmetric maps in terms of the dilatation of their quasiconformal extension. The sharpness of the estimates remains an open question and is shown to be closely…
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…
Hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under…
We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to…
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the…
We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence…
We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any…
The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and…
We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the…
In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar…