Rigidity and geometricity for surface group actions on the circle

Mann, Kathryn¹ ; Wolff, Maxime²

¹ Department of Mathematics, Cornell University, Ithaca, NY, United States
² Sorbonne Universités, UPMC Univ. Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ. Paris Diderot, Sorbonne Paris Cité, Paris, France

Geometry & topology, Tome 28 (2024) no. 5, pp. 2345-2398.

Résumé

We prove that (topologically) rigid actions of surface groups on the circle by homeomorphisms are necessarily geometric, namely, they are semiconjugate to an embedding as a cocompact lattice in a Lie group acting transitively on $S^1$. This gives the converse to a theorem of the first author; thus characterizing geometric actions as the unique isolated points in the “character space” of surface group actions on $S^1$.

Détail

DOI : 10.2140/gt.2024.28.2345

Keywords: surface group, circle homeomorphism, Euler class, rigidity, representation space

Bibliographie

[1] J Alonso, J Brum, C Rivas, Orderings and flexibility of some subgroups of Homeo+(R), J. Lond. Math. Soc. 95 (2017) 919 | DOI

[2] V A Antonov, Modeling of processes of cyclic evolution type: synchronization by a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1984) 67

[3] G Baumslag, On generalised free products, Math. Z. 78 (1962) 423 | DOI

[4] E Breuillard, T Gelander, J Souto, P Storm, Dense embeddings of surface groups, Geom. Topol. 10 (2006) 1373 | DOI

[5] M Burger, A Iozzi, A Wienhard, Higher Teichmüller spaces : from SL(2, R) to other Lie groups, from: "Handbook of Teichmüller theory, IV" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc. (2014) 539 | DOI

[6] D Calegari, A Walker, Ziggurats and rotation numbers, J. Mod. Dyn. 5 (2011) 711 | DOI

[7] A Casson, D Jungreis, Convergence groups and Seifert fibered 3–manifolds, Invent. Math. 118 (1994) 441 | DOI

[8] B Deroin, V Kleptsyn, A Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math. 199 (2007) 199 | DOI

[9] D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638 | DOI

[10] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447 | DOI

[11] É Ghys, Classe d’Euler et minimal exceptionnel, Topology 26 (1987) 93 | DOI

[12] É Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, from: "The Lefschetz centennial conference, III" (editor A Verjovsky), Contemp. Math. 58, Amer. Math. Soc. (1987) 81 | DOI

[13] É Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329

[14] W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200 | DOI

[15] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557 | DOI

[16] W M Goldman, E Z Xia, Ergodicity of mapping class group actions on SU(2)–character varieties, from: "Geometry, rigidity, and group actions" (editors B Farb, D Fisher), Univ. Chicago Press (2011) 591 | DOI

[17] A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189 | DOI

[18] D Johnson, J J Millson, Deformation spaces associated to compact hyperbolic manifolds, from: "Discrete groups in geometry and analysis" (editor R Howe), Progr. Math. 67, Birkhäuser (1987) 48 | DOI

[19] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, 54, Cambridge Univ. Press (1995) | DOI

[20] S H Kim, T Koberda, M Mj, Flexibility of group actions on the circle, 2231, Springer (2019) | DOI

[21] D Luna, Sur certaines opérations différentiables des groupes de Lie, Amer. J. Math. 97 (1975) 172 | DOI

[22] D Luna, Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble) 26 (1976) 33 | DOI

[23] K Mann, Spaces of surface group representations, Invent. Math. 201 (2015) 669 | DOI

[24] K Mann, Rigidity and flexibility of group actions on the circle, from: "Handbook of group actions, IV", Adv. Lect. Math. 41, International (2018) 705

[25] K Mann, M Wolff, A characterization of Fuchsian actions by topological rigidity, Pacific J. Math. 302 (2019) 181 | DOI

[26] J Marché, M Wolff, The modular action on PSL2(R)–characters in genus 2, Duke Math. J. 165 (2016) 371 | DOI

[27] S Matsumoto, Numerical invariants for semiconjugacy of homeomorphisms of the circle, Proc. Amer. Math. Soc. 98 (1986) 163 | DOI

[28] S Matsumoto, Some remarks on foliated S1 bundles, Invent. Math. 90 (1987) 343 | DOI

[29] S Matsumoto, Basic partitions and combinations of group actions on the circle : a new approach to a theorem of Kathryn Mann, Enseign. Math. 62 (2016) 15 | DOI

[30] J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215 | DOI

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

[32] S Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747 | DOI

[33] D Sullivan, Quasiconformal homeomorphisms and dynamics, II : Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985) 243 | DOI

[34] W P Thurston, Earthquakes in two-dimensional hyperbolic geometry, from: "Low-dimensional topology and Kleinian groups" (editor D B A Epstein), Lond. Math. Soc. Lect. Note Ser. 112, Cambridge Univ. Press (1986) 91

[35] P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1 | DOI

[36] J W Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971) 257 | DOI