We show that there exists a universal constant Kc so that every K-strongly quasiconformally homogeneous hyperbolic surface X (not equal to H2) has the property that K > Kc > 1. The constant Kc is the best possible, and is computed in terms of the diameter of the (2, 3, 7)-hyperbolic orbifold (which is the hyperbolic orbifold of smallest area.) We further show that the minimum strong homogeneity...

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume Vg,n of the moduli spaces of hyperbolic surfaces of genus g with n...

Horosphere is surface in hyperbolic space that is isometric to the Euclidean plane. In order to correctly visualize hyperbolic space we embed flat computer screen as horosphere and investigate geometry of central projection of hyperbolic space onto horosphere. We also discuss realization of hyperbolic isometries. Corresponding algorithms are implemented in Mathematica package L3toHorospere. We ...

In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.

Let Ip,v be Bourdon’s building, the unique simply connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph Kv,v. We investigate and mostly determine the set of triples (p, v, g) for which there exists a uniform lattice Γ = Γp,v,g in Aut(Ip,v) such that Γ \Ip,v is a compact orientable surface of genus g. Surp...

We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if Λ is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then Λ is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that Λ is hyperbolic. Secondly, if Λ is a nonwandering, locall...

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current st...

We study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. The study is motivated by Gromov’s famous surface subgroup question: Does every oneended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2? In [10] and [3] the authors constructed and classified all grou...

The method of common reflection surface (CRS) extend conventional stacking of seismic traces over offset to multidimensional stacking over offset-midpoint surfaces. We propose a new form of the stacking surface, derived from the analytical solution for reflection traveltime from a hyperbolic reflector. Both analytical comparisons and numerical tests show that the new approximation can be signif...