Small C1 actions of semidirect products on compact manifolds

Bonatti, Christian¹ ; Kim, Sang-hyun² ; Koberda, Thomas³ ; Triestino, Michele⁴

¹Institut de Mathematiques de Bourgogne, Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584), Dijon, France
²School of Mathematics, Korea Institute for Advanced Study (KIAS), Seoul, South Korea
³Department of Mathematics, University of Virginia, Charlottesville, VA, United States
⁴Institut de Mathématiques de Bourgogne, Universite de Bourgogne-Franche-Comté (IMB, UMR CNRS 5584), Dijon, France

Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 3183-3203

Résumé

Let T be a compact fibered 3–manifold, presented as a mapping torus of a compact, orientable surface S with monodromy ψ, and let M be a compact Riemannian manifold. Our main result is that if the induced action ψ∗ on H1(S, ℝ) has no eigenvalues on the unit circle, then there exists a neighborhood 𝒰 of the trivial action in the space of C1 actions of π1(T) on M such that any action in 𝒰 is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group H provided that the conjugation action of the cyclic group on H1(H, ℝ)≠0 has no eigenvalues of modulus one. We thus generalize a result of A McCarthy, which addressed the case of abelian-by-cyclic groups acting on compact manifolds.

Détail

Classification : 37C85, 57M60, 20E22, 37D30, 57M50, 57R35
Keywords: groups acting on manifolds, hyperbolic dynamics, fibered $3$–manifold, $C^1$–close to the identity

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045 | DOI

[2] M Aschenbrenner, S Friedl, H Wilton, 3–Manifold groups, Eur. Math. Soc. (2015) | DOI

[3] H Baik, S H Kim, T Koberda, Right-angled Artin groups in the C∞ diffeomorphism group of the real line, Israel J. Math. 213 (2016) 175 | DOI

[4] H Baik, S H Kim, T Koberda, Unsmoothable group actions on compact one-manifolds, J. Eur. Math. Soc. 21 (2019) 2333 | DOI

[5] N Bergeron, F Haglund, D T Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. 83 (2011) 431 | DOI

[6] C Bonatti, Feuilletages proches d’une fibration, 5, Soc. Brasil. Mat. (1993)

[7] 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

[8] C Bonatti, F Rezaei, Residually torsion-free nilpotent groups are C1–close to the identity, in preparation

[9] J Bowden, K Mann, C0 stability of boundary actions and inequivalent Anosov flows, preprint (2019)

[10] M Bridgeman, R Canary, F Labourie, A Sambarino, Simple root flows for Hitchin representations, Geom. Dedicata 192 (2018) 57 | DOI

[11] A Brown, D Fisher, S Hurtado, Zimmer’s conjecture : subexponential growth, measure rigidity, and strong property (T), preprint (2016)

[12] A Brown, D Fisher, S Hurtado, Zimmer’s conjecture for actions of SL(m, Z), Invent. Math. 221 (2020) 1001 | DOI

[13] M Burger, N Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. 1 (1999) 199 | DOI

[14] D Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006) 209 | DOI

[15] D Calegari, Foliations and the geometry of 3–manifolds, Oxford Univ. Press (2007)

[16] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, 9, Cambridge Univ. Press (1988) | DOI

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

[18] G Duchamp, D Krob, The lower central series of the free partially commutative group, Semigroup Forum 45 (1992) 385 | DOI

[19] B Farb, J Franks, Groups of homeomorphisms of one-manifolds, III : Nilpotent subgroups, Ergodic Theory Dynam. Systems 23 (2003) 1467 | DOI

[20] B Farb, J Franks, Groups of homeomorphisms of one-manifolds, I: Actions of nonlinear groups, from: "What’s next ?" (editor D Thurston), Princeton Univ. Press (2020) 116 | DOI

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

[22] B Farb, L Mosher, The geometry of surface-by-free groups, Geom. Funct. Anal. 12 (2002) 915 | DOI

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

[24] É Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993) 163 | DOI

[25] É Ghys, Actions de réseaux sur le cercle, Invent. Math. 137 (1999) 199 | DOI

[26] M Handel, W P Thurston, New proofs of some results of Nielsen, Adv. Math. 56 (1985) 173 | DOI

[27] S Hurder, A Katok, Differentiability, rigidity and Godbillon–Vey classes for Anosov flows, Inst. Hautes Études Sci. Publ. Math. 72 (1990) 5 | DOI

[28] S Hurtado, J Xue, A Tits alternative for surface group diffeomorphisms and abelian-by-cyclic actions on surfaces containing an Anosov diffeomorphism, preprint (2019)

[29] E Jorquera, A universal nilpotent group of C1 diffeomorphisms of the interval, Topology Appl. 159 (2012) 2115 | DOI

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

[31] S Katok, Fuchsian groups, Univ. Chicago Press (1992)

[32] S H Kim, T Koberda, Free products and the algebraic structure of diffeomorphism groups, J. Topol. 11 (2018) 1054 | DOI

[33] T Koberda, Residual properties of fibered and hyperbolic 3–manifolds, Topology Appl. 160 (2013) 875 | DOI

[34] C Maclachlan, A W Reid, The arithmetic of hyperbolic 3–manifolds, 219, Springer (2003) | DOI

[35] K Mann, M Wolff, Rigidity of mapping class group actions on S1, Geom. Topol. 24 (2020) 1211 | DOI

[36] L Marquis, J Souto, Surface groups of diffeomorphisms of the interval, Israel J. Math. 227 (2018) 379 | DOI

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

[38] S Miyoshi, On foliated circle bundles over closed orientable 3–manifolds, Comment. Math. Helv. 72 (1997) 400 | DOI

[39] J Nash, C1 isometric imbeddings, Ann. of Math. 60 (1954) 383 | DOI

[40] A Navas, Sur les rapprochements par conjugaison en dimension 1 et classe C1, Compos. Math. 150 (2014) 1183 | DOI

[41] K Parkhe, Nilpotent dynamics in dimension one: structure and smoothness, Ergodic Theory Dynam. Systems 36 (2016) 2258 | DOI

[42] K Parwani, C1 actions on the mapping class groups on the circle, Algebr. Geom. Topol. 8 (2008) 935 | DOI

[43] J F Plante, W P Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv. 51 (1976) 567 | DOI

[44] J Stallings, On fibering certain 3–manifolds, from: "Topology of –manifolds and related topics" (editor M K Fort Jr.), Prentice-Hall (1962) 95

[45] W P Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974) 347 | DOI

[46] W P Thurston, A norm for the homology of 3–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc. (1986) 99 | DOI

[47] D T Wise, The structure of groups with a quasiconvex hierarchy, book project (2011)

[48] D Witte, Arithmetic groups of higher Q–rank cannot act on 1–manifolds, Proc. Amer. Math. Soc. 122 (1994) 333 | DOI