Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname {SL}_2(\mathbb {Z})$

Hee Oh; Dale Winter

doi:10.1090/jams/849

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Compositio Mathematica ◽

10.1112/s0010437x21007600 ◽

2021 ◽

Vol 157 (12) ◽

pp. 2585-2634

Author(s):

Pratyush Sarkar ◽

Dale Winter

Keyword(s):

Asymptotic Formula ◽

Error Term ◽

Arbitrary Dimension ◽

Hyperbolic Manifolds ◽

Closed Geodesics ◽

Technical Result ◽

Exponential Mixing ◽

Spectral Bound ◽

Matrix Coefficients ◽

Convex Cocompact

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

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Twisted 6d (2, 0) SCFTs on a circle

Journal of High Energy Physics ◽

10.1007/jhep07(2021)179 ◽

2021 ◽

Vol 2021 (7) ◽

Author(s):

Zhihao Duan ◽

Kimyeong Lee ◽

June Nahmgoong ◽

Xin Wang

Keyword(s):

Lie Groups ◽

Point Of View ◽

Partition Functions ◽

Congruence Subgroups ◽

Gauge Groups ◽

Simple Lie Groups ◽

Instanton Contribution ◽

Elliptic Genera ◽

Supersymmetric Gauge ◽

The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

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The local-to-global property for Morse quasi-geodesics

Mathematische Zeitschrift ◽

10.1007/s00209-021-02811-w ◽

2021 ◽

Author(s):

Jacob Russell ◽

Davide Spriano ◽

Hung Cong Tran

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Type Theorem ◽

Hyperbolic Groups ◽

Global Property ◽

Mapping Class ◽

Relatively Hyperbolic Groups ◽

Convex Cocompact ◽

Relatively Hyperbolic ◽

Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Open Mathematics ◽

10.1515/math-2019-0115 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1631-1651

Author(s):

Ick Sun Eum ◽

Ho Yun Jung

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Singular Values ◽

Congruence Subgroups ◽

Maass Form ◽

Weakly Holomorphic Modular Forms ◽

Reciprocity Law ◽

Class Invariants ◽

Higher Genus ◽

Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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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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