Study of Schottky bundles over Riemann surfaces, proving their trivial topological type.
problem Understanding Schottky bundles and their properties over Riemann surfaces.
method Introduced and studied (strict) Schottky G-bundles, relating them to Higgs bundles and proving their trivial topological type.
result All Schottky G-bundles have trivial topological type.
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…
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…
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 paper explores non-classical Schottky groups and their properties.
problem Characterizing and understanding non-classical Schottky groups.
method Theoretical construction and analysis of infinite collections of Schottky groups.
result Construction of non-classical Schottky groups and examples.
Authors construct an example of a Schottky group of rank three.
problem Theoretical existence of non-classical Schottky groups in higher ranks.
method Provided a method to construct sufficiently complicated noded Schottky groups of any rank.
result Explicit construction of a sufficiently complicated noded Schottky group of rank three.
The article constructs Fuchsian Schottky groups with conformal boundaries.
problem Creating generalized Schottky groups with specific properties.
method Developed Fuchsian Schottky groups by including orientation-reversing isometries.
result Decomposed compact core of conformally compact Riemann surfaces into pairs of pants.
The paper describes a structural decomposition of a specific type of Schottky groups.
problem Understanding the structure of extended Z2n-Schottky groups. method Using Klein-Maskit's combination theorems.
result A structural decomposition theorem for extended Z2n-Schottky groups. The paper describes geometrically how certain groups act on surfaces.
problem Understanding the geometric structure of virtual Schottky groups.
method Geometric structural decomposition of virtual Schottky groups.
result Provides a geometrical structural decomposition for specific virtual Schottky groups.
This paper describes dihedral extended Schottky groups and their symmetries.
problem Understanding symmetries of handlebodies with Schottky structures.
method Geometrical structural description of dihedral extended Schottky groups using Klein-Maskit combination theorems.
result Sharp upper bounds for the fixed points of symmetries in handlebodies.
Study on non-classical generating sets in Fuchsian Schottky groups.
problem Estimating non-classical Schottky structure in discrete subgroups.
method Investigated Fuchsian Schottky groups with non-classical generating sets using Möbius transformations.
result Derived two non-trivial examples of Fuchsian Schottky groups with non-classical generating sets.
This paper studies connectivity of cyclic-Schottky strata in Schottky space.
problem Connectivity of cyclic-Schottky strata in Schottky space.
method Using Klein-Maskit Combination Theorems and combinatorial analysis of conjugacy classes of Schottky groups.
result Connectivity of cyclic-Schottky strata for p≥3. Characterizes groups with specific boundary properties.
problem Groups with Schottky set boundaries.
method Study relatively hyperbolic group pairs with Schottky boundaries.
result Groups with boundaries where Schottky sets have 1 or 2 component incidence graphs.
Origamis described using Schottky groups for surfaces of genus g ≥ 1.
problem Describing origamis by Schottky groups for Riemann surfaces.
method Using geometrical structural picture and Klein-Maskit combination theorems.
result Provided a geometrical structural picture of origami-Schottky groups.
We use the classical construction of Schottky groups in hyperbolic geometry to produce non-Schottky subgroups of the mapping class group.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Sharp bounds found for fixed points of symmetries in handlebodies.
problem Finding upper bounds for fixed points of symmetries in handlebodies.
method Structural description of dihedral extended Schottky groups and their fixed points.
result Sharp upper bounds for the number of fixed points components of two and three symmetries of handlebodies.
Construct Schottky subgroups for maximal representations in symmetric spaces.
problem Maximal representations of surface groups into Sp(2n, R).
method Construction of Schottky type subgroups of automorphism groups.
result Schottky subgroups correspond to maximal representations of surface groups.
Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.
problem Understanding the dimensionality of harmonic measures for random walks.
method Analyzing finite range random walks on Fuchsian Schottky groups.
result Harmonic measures have dimension strictly less than the limit set's Hausdorff dimension.
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
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…
Upper bounds on renormalized volume for Schottky groups derived from extremal lengths.
problem Comparing renormalized volumes of Schottky and Fuchsian manifolds with the same boundary.
method Bounding renormalized volume in terms of genus and extremal lengths of curves on the boundary Riemann surface.
result Upper bounds on renormalized volume for Schottky groups established.
Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).
problem Finding uniform bounds for spectral gaps of Schottky subgroups.
method Establishes explicit lower bounds for the second eigenvalue of the Laplace-Beltrami operator.
result Uniform and explicit lower bounds for the second eigenvalue of congruence coverings.
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…
New lattice extensions of Schottky groups in hyperbolic space.
problem Understanding complex translation lengths in hyperbolic manifolds.
method Produced systolic lattice extensions of Schottky subgroups.
result Density of complex translation lengths in closed hyperbolic manifolds.
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…
We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type ZΓ,Σo(λ) associated to the spinor bundle Σ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant η(D) associated to the Dirac operator D on $X…
The paper discusses methods to compute Green's function on algebraic surfaces using Schottky uniformization.
problem Computing Green's function on algebraic surfaces using Schottky uniformization.
method Investigates convergence of deformations of a formula related to Green's function.
result Provides insights into the geometric interpretation of the formula for Green's function.
In higher dimensions, Schottky spaces have unique topological properties.
problem Characterize the topology of Schottky spaces in higher dimensions.
method Analyzing the fundamental group and homotopy properties of Schottky spaces in the borderline dimension.
result In the borderline dimension, the space is simply connected but has a dense open part with fundamental group a product of cyclic groups of order two.
Schottky groups constructed from flag manifolds' partial cyclic orders.
problem Constructing Schottky groups from geometric structures.
method Using 3-hyperconvexity and partial cyclic orders on flag manifolds, constructing Schottky groups. result Schottky groups correspond to positive representations in Fock and Goncharov's sense.
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→S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω→S is a regular holomorphic cover map with Γ as it…
Researchers prove a spectral gap for Hecke covers of Schottky surfaces.
problem Proving a spectral gap for Hecke congruence covers of arithmetic Schottky surfaces.
method Using the generalized Riemann hypothesis for quadratic L-functions and properties of Schottky subgroups.
result Established a uniform and explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces.
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…
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.
Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent …
Analytic curves linked to algebraic ones via Schottky groups.
problem Moving between analytic and algebraic representations of Riemann surfaces.
method Identifying Riemann surfaces with Schottky groups and constructing families of non-hyperelliptic surfaces.
result Construction of families of non-hyperelliptic surfaces with specific properties.
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.
This paper constructs wild knots from beaded necklaces using a Schottky group.
problem Creating wild knots from beaded necklaces and studying their properties.
method Using a Schottky group generated by inversions on spheres to construct wild knots.
result The constructed wild knots are fibered if the original knot is fibered.
Identifies half-space neighborhoods of pleating rays in the Riley slice of Schottky groups.
problem Determining points in the Riley slice of Schottky groups.
method Adapting ideas from L. Keen and C. Series, identifying half-space neighborhoods of pleating rays.
result Provides a provable method to determine if a point is in the Riley slice.
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…
This paper classifies Kleinian groups with Hausdorff dimension less than 1.
problem Classifying Kleinian groups with specific Hausdorff dimensions.
method Using the result of Hou, the space of rectifiable $\G$-invariant closed curves, and properties of Schottky groups.
result Every purely loxodromic Kleinian group of Hausdorff dimension less than 1 is a classical Schottky group.
The paper classifies Kleinian groups with Hausdorff dimension less than 1.
problem Classifying Kleinian groups with specific Hausdorff dimensions.
method Using Hou's result, the paper proves that all convex cocompact Kleinian groups of Hausdorff dimension less than 1 are Schottky groups.
result The classification of convex cocompact Kleinian groups of Hausdorff dimensions less than 1.
We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to SL(2,C); and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punc…
Study compares hyperbolic and extremal lengths for shortest curves.
problem Comparing hyperbolic and extremal lengths for shortest curves.
method Lower bounds for widths of collars and upper bounds for renormalized volume of Schottky manifolds.
result Upper bounds of renormalized volume in terms of hyperbolic length of compressible curves.
Let X be a globally symmetric space of noncompact type, and $Γ\subset\Isom(X)$ a Schottky group of axial isometries. Then M:=X/Γ is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homo…
Let G∗(S,ρ) be the graph whose vertices are marked complex projective structures with holonomy ρ and whose edges are graftings from one vertex to another. If ρ is quasi-Fuchsian, a theorem of Goldman implies that G∗(S,ρ) is connected. If ρ(π1(S)) is a Schottky group Baba has shown that …
Study on smooth moduli space of Riemann surfaces with uniformization theorem.
problem Understanding the smooth moduli space of Riemann surfaces and their uniformization.
method Developed techniques of rational norm of homological marking and decomposition of probability measures.
result Closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension less than one.