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.
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. 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.
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…
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 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.
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.
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.
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.
We use the classical construction of Schottky groups in hyperbolic geometry to produce non-Schottky subgroups of the mapping class group.
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. 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.
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.
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.
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…
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…
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.
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.
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…
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…
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.
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…
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.
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…
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.
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 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 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…
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.
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.
New representations solve a gap in projective structure proof.
problem Prove every non-elementary surface group representation is a projective structure holonomy.
method Define pentagon representations and show they are non-elementary and not Schottky decomposable.
result Repair Gallo-Kapovich-Marden proof by showing pentagon representations arise as holonomies.
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…
New criterion for generating free groups in CAT(0) spaces.
problem Generating free groups from hyperbolic isometries in CAT(0) spaces.
method Criterion based on n hyperbolic isometries in a CAT(0) metric space. result Criterion extends previous results to n generators and non-complete spaces. The paper describes a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
problem Describing a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
method Constructing a hyperbolic polygon with an infinite number of sides and identifying sides in pairs with Mobius transformations.
result A precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group.
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…
Study of rank 2 Kleinian groups with specific parabolic elements.
problem Characterizing discrete representations of rank 2 free groups into PSL(2,C). method Analyzing representations sending three disjoint curves to parabolics.
result Only maximal cusp groups of infinite covolume, and new finite covolume groups.
Study of complex moduli spaces for Kleinian groups.
problem Understanding moduli spaces of Kleinian groups.
method Review and extension of existing work on the Riley slice.
result Extension of Riley slice results to new Kleinian groups.
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 …
Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
The paper studies the automorphism groups of specific 3-manifolds and finds upper bounds for their sizes.
problem Determining the size of automorphism groups of certain 3-manifolds.
method Analyzing the structure of MDC-Schottky extension groups for specific types of 3-manifolds.
result Upper bounds for the sizes of automorphism groups of specific 3-manifolds are derived and proven.