arXiv research

Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).

We use the classical construction of Schottky groups in hyperbolic geometry to produce non-Schottky subgroups of the mapping class group.

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G i…

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces wit…

It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(Ω,Γ,P:Ω\to S)$, where $Γ$ is a Schottky group with region of discontinuity $Ω$ and $P:Ω\to S$ is a regular holomorphic cover map with $Γ$ as it…

This paper studies connectivity of cyclic-Schottky strata in Schottky space.

A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their Schottky uniformizations. In this paper we provides a geometrical structural…

The paper describes a structural decomposition of a specific type of Schottky groups.

Origamis described using Schottky groups for surfaces of genus g ≥ 1.

Study on non-classical generating sets in Fuchsian Schottky groups.

New lattice extensions of Schottky groups in hyperbolic space.

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

Local-to-global principle for Morse actions on symmetric spaces.

We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomolog…

We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.

Not all Schottky groups of Moebius transformations are classical Schottky groups. In this paper we show that all Fuchsian Schottky groups are classical Schottky groups, but not necessarily on the same set of generators.

The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are "sufficiently complicated". We th…

We build examples of properly convex projective manifold $Ω/ Γ$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $Ω$ is…

The article constructs Fuchsian Schottky groups with conformal boundaries.

The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to obtain examples of non-classical Schottky …

Characterizes groups with specific boundary properties.

Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of…

Study infinite genus surfaces and Schottky groups for uniformization.

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type…

While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups $Γ$ of arithmetic groups in $PSL(2,C)^q \times PSL(2,R)^r$ with $q+r>1$ and the…

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

The paper defines infinite Schottky groups and their applications to infinite type surfaces.

A $T$-Schottky group is a discrete group of Möbius transformations whose generators identify pairs of, possibly-tangent, Jordan curves on the complex sphere, ${\hat{\IC}}$. If the curves are Euclidean circles then the group is termed classical $T$-Schottky. We describe the boundary of the space of classical $T$-Schottk…

Research examines coamenable subgroups in higher rank groups.

In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 19…

A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from pi_1(S) to th…

Greg McShane introduced a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was subsequently generalized by the authors to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical S…

The paper discusses methods to compute Green's function on algebraic surfaces using Schottky uniformization.

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodi…

In higher dimensions, Schottky spaces have unique topological properties.

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal posi…

Let $Γ$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $Γ$ into $G$ and prove that they form a domain of discontinuity for the action of $\mathrm{Out}(Γ)$. In the appendix,…

The authors exhibit pairs of infinite-volume, hyperbolic three-manifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction.

In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.

Analytic curves linked to algebraic ones via Schottky groups.

In this article we show that for any given Riemann surface $Σ$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $Σ$ in terms of the genus and the combined extremal lengths on $Σ$ of $(g-1)$ disjoint, non-homotopic, simple closed comp…

This paper constructs wild knots from beaded necklaces using a Schottky group.

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary K…

Identifies half-space neighborhoods of pleating rays in the Riley slice of Schottky groups.

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient cond…

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…