New findings on infinite subgroups in negatively curved spaces.
problem Characterizing infinite discrete isometry subgroups in negatively pinched Hadamard manifolds.
method Generalization of Bonahon's characterization to negatively pinched Hadamard manifolds.
result Every geometrically infinite isometry subgroup has a continuum of nonconical limit points.
3-manifolds have covers with infinitely many ideal triangulations.
problem Proving the existence of infinitely many geometric ideal triangulations in certain 3-manifolds.
method Using separability of peripheral subgroups and conjugacy separability theorems.
result Every cusped hyperbolic 3-manifold has a cover with infinitely many geometric ideal triangulations.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
The paper extends geometric finiteness to discrete subgroups of negatively pinched Hadamard manifolds.
problem Characterizing geometrically infinite discrete subgroups of negatively pinched Hadamard manifolds.
method Generalizing Bonahon's characterization and proving a theorem of Bishop's extension.
result Every discrete geometrically infinite isometry subgroup has a set of nonconical limit points of cardinality continuum.
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.
Study fundamental groups of geometric transformation groups using loop spaces.
problem Understanding fundamental groups of geometric transformation groups.
method Use differential forms on loop spaces to prove infinite fundamental groups.
result Proves infinite fundamental groups for specific geometric transformation groups.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
Paper studies Hausdorff dimension of limit sets for Kleinian groups.
problem Understanding Hausdorff dimension of limit sets for Kleinian groups.
method Constructs geometrically infinite Fuchsian groups and proves properties for finitely generated groups.
result Hausdorff dimension of nonconical limit set equals zero for some groups.
New surfaces show horocyclic flow isn't always minimal.
problem Complex dynamics on infinite fineness surfaces.
method Construction of infinite hyperbolic surfaces.
result Horocyclic flow is not minimal on infinite fineness surfaces.
This is a summary of the material for 3 lectures on geometrically finite and infinite Kleinian groups delivered by the author at a workshop held at Tata Institute of Fundamental Research in April 2014.
We prove a partial generalization of Bonahon's tameness result to surfaces inside irreducible 3-manifolds with hyperbolic fundamental group. Bonahon's result states that geometrically infinite ends of freely indecomposable hyperbolic 3-manifolds are simply degenerate. It is easy to see that a geometrically infinite end…
Let X0 be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric …
Geodesics grow infinitely in certain Finsler manifolds.
problem Understanding the growth of geodesic chords in Finsler manifolds.
method Analyzing properties of forward complete Finsler manifolds with infinite fundamental group.
result Geodesics grow infinitely in specific Finsler manifolds.
Develops geometric quantum mechanics in infinite dimensions.
problem Quantum dynamics in infinite-dimensional spaces.
method Tulczyjew triple concept for Lagrangian formalism.
result Self-adjoint operators as Lagrangian submanifolds.
In this article we give the realization of the Klein's Program for geometrical structures (Riemannian spaces and fiber bundles with connection) with arbitrary variable curvature within the framework of infinite deformed groups. These groups generalize gauge groups to the case of nontrivial action on the base space of b…
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
The study reveals conditions for infinite closed geodesics on specific surfaces.
problem Conditions for infinite closed geodesics on complete surfaces.
method Analyzes geometric and homological properties of closed geodesics on cylinders and planes.
result Proves that complete cylinders with isolated geodesics have zero, one, or infinitely many homologically visible geodesics.
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter a…
The study finds infinitely many distinct prime geodesics on Riemannian manifolds.
problem Existence of infinitely many geometrically distinct, non-constant prime closed geodesics.
method Handle decomposition of free loop space and cellular homology.
result Energy functional has infinitely many non-degenerate critical points.
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…
Any action of a group Γ on H3 by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to Γ. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends…
This article presents some methods to control the bottom of the spectrum of the Laplacian λ0 on hyperbolic surfaces with infinite volume. Our first result bounds the λ0 of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built …
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non-hR minimal closures. Geometric approach solves Euler equations with random forces.
problem Solving Euler equations with stochastic forcing.
method Infinite-dimensional geometric approach, combining stochastic analysis and Sobolev mappings.
result Local existence and uniqueness of strong solutions.
Workshop on shape analysis discusses new research directions.
problem No specific problem stated; focus on new directions in shape analysis.
method Discussion and collaboration among researchers.
result Promising new directions in shape analysis were discussed.
New constraints found for algebro-geometric subgroups of mapping class groups.
problem Constraints for algebro-geometric subgroups of mapping class groups.
method Using deep work of Gibney, Keel, and Morrison, constraints on the Shafarevich morphism are derived to prove the infinite restriction of certain representations.
result Most Reshetikhin-Turaev representations of the mapping class group restrict to infinite representations on algebro-geometric subgroups when the genus is at least 3.
This paper opens the series of articles supplemental to the series (hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003), which also lies in lines of general ideology exposed in the review (mp_arc/96-477). The main purpose of the activity, which has its origin and motivation presumably in the auth…
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In…
Geometric framework for Newton's equations on diffeomorphism groups.
problem Modeling fluid dynamics and related systems on geometric spaces.
method Geodesic approach and infinite-dimensional information geometry.
result Unified framework for various fluid dynamics equations.
We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.
problem Bounding eigenvalues of geometrically finite hyperbolic manifolds of infinite volume.
method Provided a lower bound on the kth eigenvalue of the Laplace-Beltrami operator by the kth eigenvalue of a neighborhood of the thick part of the convex core.
result Recovered a theorem bounding the bottom eigenvalue from below by a specific formula involving the volume of the 1-neighborhood of the convex core.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
Study on geodesics on connected sums of manifolds, showing infinite number of distinct closed geodesics.
problem Counting geometrically distinct closed geodesics on connected sums of manifolds.
method Analyzing the asymptotics of the number of closed geodesics on Riemannian or Finsler metrics.
result Generic Riemannian metrics on compact 3-manifolds have infinitely many geometrically distinct closed geodesics.
Defines an equivariant cobordism category for finite groups.
problem Understanding geometric structures with group actions.
method Introduces a new category for (d−1)-dimensional closed smooth G-manifolds and their cobordisms. result Identifies the homotopy type of the classifying space of the equivariant cobordism category as fixed points of an infinite loop space.
A 3D space of hyperbolic manifolds is connected but not path-connected.
problem Proving connectivity and non-path-connectedness of framed hyperbolic 3-manifolds.
method Two proofs using density theorems for Kleinian groups, constructing dense sets of framings, and discussing paths.
result The space of framed infinite volume hyperbolic 3-manifolds is not path-connected.
We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform ir…
Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.
Paper constructs an infinite 3-7 surface in 3D space.
problem Can an infinite 3-7 surface be embedded in Euclidean space?
method Constructs an immersed, but not embedded, infinite 3-7 surface in R^3.
result Realizes an infinite 3-7 surface as a cover of Klein's quartic.
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.
We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…
The study finds infinite geodesics on manifolds with specific homotopy group properties.
problem Existence of non-contractible closed geodesics on manifolds with certain homotopy group properties.
method Analyzes the homotopy groups and their action on the fundamental group to prove the existence of infinitely many closed geodesics.
result Infinitely many geometrically distinct closed geodesics exist under specific conditions on homotopy groups.
Geometric model for flag waving motion.
problem Modeling the motion of a physical flag.
method Isometric immersion of a square into 3D space with boundary conditions.
result The space of flags is an infinite dimensional manifold.