Ratner’s rigidity theorem for geometrically finite Fuchsian groups

2016 ◽

Vol 12 (06) ◽

pp. 1663-1668

Author(s):

Joshua S. Friedman

Keyword(s):

Lower Bound ◽

Fixed Points ◽

Fuchsian Group ◽

Closed Geodesic ◽

Universal Constant ◽

Fuchsian Groups ◽

Hyperbolic Distance ◽

Minimal Distance ◽

Geometrically Finite ◽

Upper Half Plane

Let [Formula: see text] be a geometrically finite Fuchsian group acting on the upper half-plane [Formula: see text]. Let [Formula: see text] denote the set of elliptic fixed points of [Formula: see text] in [Formula: see text]. We give a lower bound on the minimal hyperbolic distance between points in [Formula: see text]. Our bound depends on a universal constant and the length of the smallest closed geodesic on [Formula: see text].

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Profinite rigidity for twisted Alexander polynomials

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0014 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Jun Ueki

Keyword(s):

Rigidity Theorem ◽

Homology Groups ◽

Alexander Polynomials ◽

Twisted Homology ◽

Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

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Geodesic planes in geometrically finite acylindrical -manifolds

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.19 ◽

2021 ◽

pp. 1-40

Author(s):

YVES BENOIST ◽

HEE OH

Keyword(s):

Closed Geodesic ◽

Duke Math ◽

Geometrically Finite ◽

Convex Cocompact ◽

Invent Math ◽

Convex Core

Abstract Let M be a geometrically finite acylindrical hyperbolic $3$ -manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math.209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$ -manifold $M_0$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$ . We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

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