scholarly journals The ending lamination conjecture for hyperbolic three-manifolds with slender end-invariants

2006 ◽  
Vol 225 (2) ◽  
pp. 231-241
Author(s):  
Richard Allen Evans
Keyword(s):  
Hyperbolic Three Manifolds ◽  
Ending Lamination

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

scholarly journals Local Mixing and Invariant Measures for Horospherical Subgroups on Abelian Covers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6036-6088
Author(s):  
Hee Oh ◽  
Wenyu Pan
Keyword(s):  
Lie Group ◽  
Invariant Measures ◽  
Dense Subgroup ◽  
Abelian Cover ◽  
Rank One ◽  
Hyperbolic Three Manifolds ◽  
Abelian Covers ◽  
Prime Geodesic Theorem ◽  
Convex Cocompact ◽  
Frame Flow

Abstract Abelian covers of hyperbolic three-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic three-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space Γ0∖G where G is a rank one simple Lie group and Γ0 < G is a convex cocompact Zariski dense subgroup.

scholarly journals Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

2020 ◽  
Vol 2020 (758) ◽  
pp. 1-66
Author(s):  
Jeffrey Brock ◽  
Christopher Leininger ◽  
Babak Modami ◽  
Kasra Rafi
Keyword(s):  
Closed Surface ◽  
Maximal Dimension ◽  
Curve Complex ◽  
Limit Sets ◽  
Ergodic Measures ◽  
Finite Set ◽  
Geodesic Rays ◽  
Thurston Boundary ◽  
Ending Lamination ◽  
Gromov Boundary

AbstractGiven a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3}, for example.We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

scholarly journals The ending lamination space of the five-punctured sphere is the Nöbeling curve

2011 ◽  
Vol 84 (1) ◽  
pp. 103-119 ◽  
Author(s):  
Sebastian Hensel ◽  
Piotr Przytycki
Keyword(s):  
Ending Lamination

scholarly journals On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds

2017 ◽  
Vol 27 (5) ◽  
pp. 1106-1160 ◽  
Author(s):  
Francesco Bonsante ◽  
Andrea Seppi ◽  
Andrea Tamburelli
Keyword(s):  
De Sitter ◽  
Globally Hyperbolic ◽  
Hyperbolic Three Manifolds

scholarly journals Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1762
Author(s):  
Dženan Gušić
Keyword(s):  
Riemann Surfaces ◽  
Symmetric Spaces ◽  
Hyperbolic Manifolds ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Hyperbolic Three Manifolds ◽  
Positive Roots ◽  
Prime Geodesic Theorem ◽  
Compact Riemann Surfaces ◽  
Negative Sectional Curvature

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρηlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

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