The asymmetry of Thurston’s earthquake flow

Arana-Herrera, Francisco¹ ; Wright, Alex²

¹ Department of Mathematics, University of Maryland, College Park, MD, United States
² Department of Mathematics, University of Michigan, Ann Arbor, MI, United States

Geometry & topology, Tome 28 (2024) no. 5, pp. 2125-2144.

Résumé

We show that Thurston’s earthquake flow is strongly asymmetric in the sense that its normalizer is as small as possible inside the group of orbifold automorphisms of the bundle of measured geodesic laminations over moduli space. (At the level of Teichmüller space, such automorphisms correspond to homeomorphisms that are equivariant with respect to an automorphism of the mapping class group.) It follows that the earthquake flow does not extend to an $\mathrm{SL}(2,ℝ)$–action of orbifold automorphisms and does not admit continuous renormalization self-symmetries. In particular, it is not conjugate to the Teichmüller horocycle flow via an orbifold map. This contrasts with a number of previous results, most notably Mirzakhani’s theorem that the earthquake and Teichmüller horocycle flows are measurably conjugate.

Détail

DOI : 10.2140/gt.2024.28.2125

Keywords: earthquake flow, normalizer, centralizer, dynamics, asymmetry, hyperbolic geometry, Teichmüller theory

Bibliographie

[1] F Arana-Herrera, Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces, Geom. Dedicata 210 (2021) 65 | DOI

[2] F Arana-Herrera, Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes, Geom. Topol. 26 (2022) 1291 | DOI

[3] A Avila, S Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math. 178 (2013) 385 | DOI

[4] A Avila, M J Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials, Comment. Math. Helv. 87 (2012) 589 | DOI

[5] A Avila, S Gouëzel, J C Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006) 143 | DOI

[6] L Bakker, T Fisher, Open sets of diffeomorphisms with trivial centralizer in the C1 topology, Nonlinearity 27 (2014) 2869 | DOI

[7] P Berk, K Frączek, T De La Rue, On typicality of translation flows which are disjoint with their inverse, J. Inst. Math. Jussieu 19 (2020) 1677 | DOI

[8] C Bonatti, I Monteverde, A Navas, C Rivas, Rigidity for C1 actions on the interval arising from hyperbolicity, I : Solvable groups, Math. Z. 286 (2017) 919 | DOI

[9] W Bonomo, P Varandas, A criterion for the triviality of the centralizer for vector fields and applications, J. Differential Equations 267 (2019) 1748 | DOI

[10] L Burslem, A Wilkinson, Global rigidity of solvable group actions on S1, Geom. Topol. 8 (2004) 877 | DOI

[11] A Calderon, J Farre, Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow, Geom. Topol. 28 (2024) 1995 | DOI

[12] V Erlandsson, J Souto, Counting curves on orbifolds, Trans. Lond. Math. Soc. 9 (2022) 56 | DOI

[13] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[14] K Frączek, M Lemańczyk, On the self-similarity problem for ergodic flows, Proc. Lond. Math. Soc. 99 (2009) 658 | DOI

[15] K Frączek, J Kułaga-Przymus, M Lemańczyk, Non-reversibility and self-joinings of higher orders for ergodic flows, J. Anal. Math. 122 (2014) 163 | DOI

[16] S W Fu, Cusp excursions for the earthquake flow on the once-punctured torus, Conform. Geom. Dyn. 23 (2019) 251 | DOI

[17] É Ghys, Actions localement libres du groupe affine, Invent. Math. 82 (1985) 479 | DOI

[18] É Ghys, A Verjovsky, Locally free holomorphic actions of the complex affine group, from: "Geometric study of foliations" (editors T Mizutani, K Masuda, S Matsumoto, T Inaba, T Tsuboi, Y Mitsumatsu), World Sci. (1994) 201

[19] N Guelman, I Liousse, C1–actions of Baumslag–Solitar groups on S1, Algebr. Geom. Topol. 11 (2011) 1701 | DOI

[20] N Guelman, I Liousse, Actions of Baumslag–Solitar groups on surfaces, Discrete Contin. Dyn. Syst. 33 (2013) 1945 | DOI

[21] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221 | DOI

[22] N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Int. Math. Res. Not. 1997 (1997) 651 | DOI

[23] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235 | DOI

[24] M Lipnowski, A Wright, Towards optimal spectral gaps in large genus, Ann. Probab. 52 (2024) 545 | DOI

[25] M Liu, Length statistics of random multicurves on closed hyperbolic surfaces, Groups Geom. Dyn. 16 (2022) 437 | DOI

[26] S Lu, W Su, Counting mapping class group orbits under shearing coordinates, Geom. Dedicata 216 (2022) 16 | DOI

[27] H Masur, The Teichmüller flow is Hamiltonian, Proc. Amer. Math. Soc. 123 (1995) 3739 | DOI

[28] A E Mccarthy, Rigidity of trivial actions of abelian-by-cyclic groups, Proc. Amer. Math. Soc. 138 (2010) 1395 | DOI

[29] C T Mcmullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283 | DOI

[30] Y Minsky, B Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math. 552 (2002) 131 | DOI

[31] M Mirzakhani, Random hyperbolic surfaces and measured laminations, from: "In the tradition of Ahlfors–Bers, IV" (editors D Canary, J Gilman, J Heinonen, H Masur), Contemp. Math. 432, Amer. Math. Soc. (2007) 179 | DOI

[32] M Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007) 179 | DOI

[33] M Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. 2008 (2008) | DOI

[34] M Mirzakhani, Counting mapping class group orbits on hyperbolic surfaces, preprint (2016)

[35] A Navas, Groups of circle diffeomorphisms, Univ. of Chicago Press (2011) | DOI

[36] A Navas, Group actions on 1–manifolds : a list of very concrete open questions, from: "Proceedings of the International Congress of Mathematicians, III" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 2035

[37] D Obata, Symmetries of vector fields: the diffeomorphism centralizer, Discrete Contin. Dyn. Syst. 41 (2021) 4943 | DOI

[38] R C Penner, J L Harer, Combinatorics of train tracks, 125, Princeton Univ. Press (1992) | DOI

[39] M Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems 7 (1987) 267 | DOI

[40] J Smillie, B Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004) 249 | DOI

[41] Y Sözen, F Bonahon, The Weil–Petersson and Thurston symplectic forms, Duke Math. J. 108 (2001) 581 | DOI

[42] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[43] W P Thurston, Minimal stretch maps between hyperbolic surfaces, preprint (1998)

[44] A Wilkinson, Conservative partially hyperbolic dynamics, from: "Proceedings of the International Congress of Mathematicians, III" (editors R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, P Gastesi), Hindustan (2010) 1816

[45] A Wilkinson, J Xue, Rigidity of some abelian-by-cyclic solvable group actions on TN, Comm. Math. Phys. 376 (2020) 1223 | DOI

[46] A Wright, A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces, Bull. Amer. Math. Soc. 57 (2020) 359 | DOI

[47] A Wright, Mirzakhani’s work on earthquake flow, from: "Teichmüller theory and dynamics" (editors P Dehornoy, E Lanneau), Panor. Synth. 58, Soc. Math. France (2022) 101