VOLUME BOUND FOR THE CANONICAL LIFT COMPLEMENT OF A RANDOM GEODESIC

CREMASCHI, Tommaso; Krifka, Yannick; MARTINEZ GRANADO, Didac; Pallete, Franco Vargas

doi:10.1090/btran/152

Download

Article (Scientific journals)

VOLUME BOUND FOR THE CANONICAL LIFT COMPLEMENT OF A RANDOM GEODESIC

CREMASCHI, Tommaso; Krifka, Yannick; MARTINEZ GRANADO, Didac et al.

2023 • In Transactions of the American Mathematical Society. Series B, 10 (28), p. 988 - 1038

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/10993/67009

DOI
10.1090/btran/152

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

volume_random_geodesic_TAMS_OA.pdf

Author postprint (1.16 MB)

Download

All documents in ORBilu are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Mathematics (miscellaneous)

Abstract :

[en] Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve’s canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length for a collection of curves with asymptotic density one. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.

Disciplines :

Mathematics

Author, co-author :

CREMASCHI, Tommaso ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Mathematics > Team Jean-Marc SCHLENKER

Krifka, Yannick ; Max-Planck-Institut für Mathematik, Bonn, Germany

MARTINEZ GRANADO, Didac ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Pallete, Franco Vargas; Department of Mathematics, Yale University, New Haven, United States

External co-authors :

yes

Language :

English

Title :

VOLUME BOUND FOR THE CANONICAL LIFT COMPLEMENT OF A RANDOM GEODESIC

Publication date :

2023

Journal title :

Transactions of the American Mathematical Society. Series B

eISSN :

2330-0000

Publisher :

American Mathematical Society

Volume :

Issue :

Pages :

988 - 1038

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.ams.org/btran/2023-10-28/S2330-0000-2023-00152-2/S2330-0000-2023-00152-2.pdf

Funders :

NSF - National Science Foundation

Funding text :

Received by the editors March 28, 2022, and, in revised form, January 13, 2023. 2020 Mathematics Subject Classification. Primary 57M50, 57K32, 57K20, 37A25. This work was supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester. The fourth author\u2019s research was supported by NSF grant DMS-2001997.

Available on ORBilu :

since 30 December 2025

Statistics

Number of views

21 (0 by Unilu)

Number of downloads

3 (0 by Unilu)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

[AL17] Javier Aramayona and Christopher J. Leininger, Hyperbolic structures on surfaces and geodesic currents, Algorithmic and geometric topics around free groups and automorphisms, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2017, pp. 111–149. MR3752040
[AST07] Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J.Amer.Math.Soc.20 (2007), no. 4, 1053–1077, DOI 10.1090/S0894-0347-07-00564-4. With an appendix by Nathan Dunfield. MR2328715
[Aub98] Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998, DOI 10.1007/978-3-662-130063. MR1636569
[Bab02] Martine Babillot, On the mixing property for hyperbolic systems, IsraelJ.Math.129 (2002), 61–76, DOI 10.1007/BF02773153. MR1910932
[Bas13] Ara Basmajian, Universal length bounds for non-simple closed geodesics on hyperbolic surfaces, J. Topol. 6 (2013), no. 2, 513–524, DOI 10.1112/jtopol/jtt005. MR3065183
[BEG17] Michael Björklund, Manfred Einsiedler, and Alexander Gorodnik, Quantitative multiple mixing, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 5, 1475–1529, DOI 10.4171/jems/949. MR4081727
[BIPP21] M. Burger, A. Iozzi, A. Parreau, and M. B. Pozzetti, Currents, systoles, and compactifications of character varieties, Proc. Lond. Math. Soc. (3) 123 (2021), no. 6, 565–596, DOI 10.1112/plms.12419. MR4368682
[Bon86] Francis Bonahon, Bouts des variétés hyperboliques de dimension 3 (French), Ann. of Math. (2) 124 (1986), no. 1, 71–158, DOI 10.2307/1971388. MR847953
[Bon88] Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162, DOI 10.1007/BF01393996. MR931208
[BP92] Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry,Universitext, Springer-Verlag, Berlin, 1992, DOI 10.1007/978-3-642-58158-8. MR1219310
[BPS17] Ara Basmajian, Hugo Parlier, and Juan Souto, Geometric filling curves on surfaces, Bull. Lond. Math. Soc. 49 (2017), no. 4, 660–669, DOI 10.1112/blms.12057. MR3725486
[BPS19a] Maxime Bergeron, Tali Pinsky, and Lior Silberman, An upper bound for the volumes of complements of periodic geodesics, Int. Math. Res. Not. IMRN 15 (2019), 4707–4729, DOI 10.1093/imrn/rnx231. MR3988666
[BPS19b] Alex Brandts, Tali Pinsky, and Lior Silberman, Volumes of hyperbolic three-manifolds associated with modular links, Symmetry 11 (2019), no. 10, 1206 https://www.mdpi. com/2073-8994/11/10/1206.
[Bri04] Martin Bridgeman, Random geodesics, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 97–107, DOI 10.1090/conm/355/06447. MR2145058
[Bri11] Martin Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom.Topol.15 (2011), no. 2, 707–733, DOI 10.2140/gt.2011.15.707. MR2800364
[Bur90] Marc Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J. 61 (1990), no. 3, 779–803, DOI 10.1215/S0012-7094-90-06129-0. MR1084459
[Cal] Danny Calegary, https://lamington.wordpress.com/2012/02/11/fillinggeodesics-and-hyperbolic-complements/.
[CRM20] Tommaso Cremaschi and José A.Rodríguez-Migueles, Hyperbolicity of link complements in Seifert-fibered spaces, Algebr. Geom. Topol. 20 (2020), no. 7, 3561–3588, DOI 10.2140/agt.2020.20.3561. MR4194288
[CRMY22] Tommaso Cremaschi, José Andrés Rodriguŕz-Migueles, and Andrew Yarmola, On volumes and filling collections of multicurves, J. Topol. 15 (2022), no. 3, 1107–1153, DOI
10.1112/topo.12246. MR4442684 [Dal11] Françoise Dal’Bo, Geodesic and horocyclic trajectories, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Translated from the 2007 French original, DOI 10.1007/978-0-85729-073-1. MR2766419
[Duk88] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90, DOI 10.1007/BF01393993. MR931205
[Ebe96] Patrick B. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. MR1441541
[FH13] Patrick Foulon and Boris Hasselblatt, Contact Anosov flows on hyperbolic 3manifolds, Geom.Topol.17 (2013), no. 2, 1225–1252, DOI 10.2140/gt.2013.17.1225. MR3070525
[FH19] Todd Fisher and Boris Hasselblatt, Hyperbolic flows, Zurich Lectures in Advanced
Mathematics, EMS Publishing House, Berlin, 2019, DOI 10.4171/200. MR3972204 [Ghy07] Étienne Ghys, Knots and dynamics, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 247–277, DOI 10.4171/022-1/11. MR2334193
[Hat07] Allen Hatcher, Basic notes on 3-manifolds, 2007, http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf.
[Hem76] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR0415619
[Hub60] Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II (German), Math. Ann. 142 (1960/61), 385–398, DOI 10.1007/BF01451031. MR126549
[Jac80] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR565450
[KH95] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza, DOI 10.1017/CBO9780511809187. MR1326374
[Mar04] Grigoriy A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska, DOI 10.1007/978-3-662-09070-1. MR2035655
[RM20] José Andrés Rodríguez-Migueles, A lower bound for the volumes of complements of periodic geodesics, J. Lond. Math. Soc. (2) 102 (2020), no. 2, 695–721, DOI 10.1112/jlms.12333. MR4171431
[RM21] José Andrés Rodríguez-Migueles, Periods of continued fractions and volumes of modular knots complements, 2021, arXiv:2008.12436.
[PP13] Jouni Parkkonen and Frédéric Paulin, Counting arcs in negative curvature, Geometry, topology, and dynamics in negative curvature, London Math. Soc. Lecture Note Ser., vol. 425, Cambridge Univ. Press, Cambridge, 2016, pp. 289–344. MR3497264
[PP17] Jouni Parkkonen and Frédéric Paulin, Counting common perpendicular arcs in negative curvature, Ergodic Theory Dynam. Systems 37 (2017), no. 3, 900–938, DOI 10.1017/etds.2015.77. MR3628925
[Sap17] Jenya Sapir, A Birman-series type result for geodesics with infinitely many selfintersections, Trans. Amer. Math. Soc. 374 (2021), no. 11, 7553–7568, DOI 10.1090/tran/8108. MR4328675
[Wal82] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR648108
[Wil14] Amie Wilkinson, Lectures on marked length spectrum rigidity, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 283–324, DOI 10.1090/pcms/021/09. MR3329731