Related papers: Hyperbolic Lambert Quadrilaterals and Quasiconform…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to…
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…
This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…
The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^d + c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic…
We classify all negatively curved $\R^n \rtimes \R$ up to quasiisometry. We show that all quasiisometries between such manifolds (except when they are biLipschitz to the real hyperbolic spaces) are almost similarities. We prove these…
A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…
Based on a notion by Gray and Kambites of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups, we will construct (under a small additional geometric assumption) a boundary based on quasi-geodesic rays and anti-rays…
It is well-known that the $L^2$ metric on the moduli space of hyperbolic monopoles, defined using the Coulomb gauge-fixing condition, diverges. This article shows that an alternative gauge-fixing condition inspired by supersymmetry cures…
We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the $l^1$-homology of a word-hyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a…
We show that every cubic Blaschke product has a unique hyperbolic inflection point in the unit disk and, moreover, this point lies at the hyperbolic midpoint of the two critical points. Using this structure result for cubic Blaschke…
This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous…
This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this…
In this paper, we establish effective equidistribution of transverse intersection points between properly immersed totally geodesic submanifolds of complementary dimensions in a finite-volume hyperbolic manifold with respect to the…
We investigate lower asymptotic bounds of number variances for invariant locally square-integrable random measures on Euclidean and real hyperbolic spaces. In the Euclidean case we show that there are subsequences of radii for which the…
We establish sharp bounds for simultaneous local rotation and H\"older-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates…
In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case…
Hyperbolic geometry plays an important role within function theory of the disk. For example, via the Schwarz-Pick Lemma, the isometries of the unit disk $\mathbb D$ with respect to this geometry are the conformal self-maps of $\mathbb D$.…
This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with…
We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical…