## Diffeomorphism groups of critical regularity.(English)Zbl 1446.57020

For a circle or a compact real interval $$M$$, let $$\mathrm{Diff}_+^\alpha(M)=\mathrm{Diff}_+^{k + \tau}(M)$$ denote the group of orientation preserving $$C^k$$ diffeomorphisms of $$M$$ whose $$k$$th derivatives are Hölder continuous with exponent $$\tau=\alpha-k$$ where $$k$$ denotes the integral part of $$\alpha$$. As the authors note, the purpose of the present paper is to study the algebraic structure of finitely generated subgroups of $$\mathrm{Diff}_+^\alpha(M)$$ for varying $$\alpha$$; in particular, the authors give the first construction of finitely generated groups and of countable simple groups in $$\mathrm{Diff}_+^\alpha(M)$$ which are not contained in the union $$\bigcup_{\beta > \alpha}\mathrm{Diff}_+^\beta(M)$$. More generally, the main result states that there is a continuum of isomorphism types of finitely generated subgroups $$G$$ of $$\mathrm{Diff}_+^\alpha(M)$$ (with simple commutator subgroups) such that $$G$$ admits no injective homomorphisms into $$\bigcup_{\beta > \alpha}\mathrm{Diff}_+^\beta(M)$$; dually, they prove that there is a continuum of isomorphism types of finitely generated subgroups $$G$$ of $$\bigcap_{\beta < \alpha}\mathrm{Diff}_+^\beta(M)$$ such that $$G$$ admits no injective homomorphism into $$\mathrm{Diff}_+^\alpha(M)$$. Some applications to smoothability of codimension one foliations are given; also, the class of finitely generated subgroups of $$\mathrm{Diff}_+^1(M)$$ is not closed under taking finite free products.

### MSC:

 57M60 Group actions on manifolds and cell complexes in low dimensions 20F36 Braid groups; Artin groups 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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### References:

 [1] Bader, U.; Furman, A.; Gelander, T.; Monod, N., Property (T) and rigidity for actions on Banach spaces, Acta Math., 198, 1, 57-105 (2007) · Zbl 1162.22005 [2] Baik, H.; Kim, S.; Koberda, T., Unsmoothable group actions on compact one-manifolds, J. Eur. Math. Soc. (JEMS), 21, 8, 2333-2353 (2019) · Zbl 1454.20074 [3] Baik, H.; Kim, S.; Koberda, T., Right-angled Artin groups in the $$C^\infty$$ diffeomorphism group of the real line, Isr. J. Math., 213, 1, 175-182 (2016) · Zbl 1398.20045 [4] Banyaga, A., The Structure of Classical Diffeomorphism Groups, Mathematics and its Applications (1997), Dordrecht: Kluwer Academic Publishers Group, Dordrecht [5] Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin (1976), Grundlehren der Mathematischen Wissenschaften, No. 223 · Zbl 0344.46071 [6] Bergman, GM, Right orderable groups that are not locally indicable, Pac. J. Math., 147, 2, 243-248 (1991) · Zbl 0677.06007 [7] Bonatti, C.; Lodha, Y.; Triestino, M., Hyperbolicity as an obstruction to smoothability for one-dimensional actions, Geom. Topol., 23, 4, 1841-1876 (2019) · Zbl 1428.37029 [8] Bonatti, C.; Monteverde, I.; Navas, A.; Rivas, C., Rigidity for $$C^1$$ actions on the interval arising from hyperbolicity I: solvable groups, Math. Z., 286, 3-4, 919-949 (2017) · Zbl 1433.37030 [9] Bowditch, BH; Sakuma, M., The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface, Groups Geom. Dyn., 12, 2, 703-719 (2018) · Zbl 1456.20042 [10] Brin, MG, Higher dimensional Thompson groups, Geom. Dedicata, 108, 163-192 (2004) · Zbl 1136.20025 [11] Brin, MG; Squier, CC, Groups of piecewise linear homeomorphisms of the real line, Invent. Math., 79, 3, 485-498 (1985) · Zbl 0563.57022 [12] Burger, M.; Monod, N., Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS), 1, 2, 199-235 (1999) · Zbl 0932.22008 [13] Burger, M.; Mozes, S., Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris Sér. I Math., 324, 7, 747-752 (1997) · Zbl 0966.20013 [14] Burillo, J.: Thompson’s group F (preprint) (2016) [15] Calegari, D., Dynamical forcing of circular groups, Trans. Am. Math. Soc., 358, 8, 3473-3491 (2006) · Zbl 1134.37015 [16] Calegari, D., Nonsmoothable, locally indicable group actions on the interval, Algebr. Geom. Topol., 8, 1, 609-613 (2008) · Zbl 1154.37015 [17] Camacho, C., Neto, A.L.: Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA (1985), Translated from the Portuguese by Sue E. Goodman · Zbl 0568.57002 [18] Candel, A., Conlon, L.: Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI (2000) · Zbl 0936.57001 [19] Cannon, JW; Floyd, WJ; Parry, WR, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), 42, 3-4, 215-256 (1996) · Zbl 0880.20027 [20] Cantwell, J.; Conlon, L., Nonexponential leaves at finite level, Trans. Am. Math. Soc., 269, 2, 637-661 (1982) · Zbl 0487.57009 [21] Cantwell, J.; Conlon, L., Smoothability of proper foliations, Ann. Inst. Fourier (Grenoble), 38, 3, 219-244 (1988) · Zbl 0644.57013 [22] Castro, G.; Jorquera, E.; Navas, A., Sharp regularity for certain nilpotent group actions on the interval, Math. Ann., 359, 1-2, 101-152 (2014) · Zbl 1310.37020 [23] Denjoy, A., Sur la continuité des fonctions analytiques singulières, Bull. Soc. Math. France, 60, 27-105 (1932) · JFM 58.1077.04 [24] Deroin, B.; Kleptsyn, V.; Navas, A., Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199, 2, 199-262 (2007) · Zbl 1139.37025 [25] Epstein, DBA, The simplicity of certain groups of homeomorphisms, Compos. Math., 22, 165-173 (1970) · Zbl 0205.28201 [26] Evans, LC, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics (2010), Providence, RI: American Mathematical Society, Providence, RI [27] Farb, B., Franks, J.: Groups of homeomorphisms of one-manifolds, I: actions of nonlinear groups, ArXiv Mathematics e-prints (2001) [28] Farb, B.; Franks, J., Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergod. Theory Dyn. Syst., 23, 5, 1467-1484 (2003) · Zbl 1037.37020 [29] Ghys, É., Actions de réseaux sur le cercle, Invent. Math., 137, 1, 199-231 (1999) · Zbl 0995.57006 [30] Ghys, É.; Sergiescu, V., Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62, 2, 185-239 (1987) · Zbl 0647.58009 [31] Goodman, SE, Closed leaves in foliations of codimension one, Comment. Math. Helv., 50, 3, 383-388 (1975) · Zbl 0318.57027 [32] Guelman, N.; Liousse, I., $$C^1$$-actions of Baumslag-Solitar groups on $$S^1$$, Algebr. Geom. Topol., 11, 3, 1701-1707 (2011) · Zbl 1221.37048 [33] Handel, M.; Thurston, WP, New proofs of some results of Nielsen, Adv. Math., 56, 2, 173-191 (1985) · Zbl 0584.57007 [34] Hölder, O.: The axioms of quantity and the theory of measurement. J. Math. Psych. 40 (1996), no. 3, 235-252, Translated from the 1901 German original and with notes by Joel Michell and Catherine Ernst, With an introduction by Michell · Zbl 0889.92035 [35] Hurtado, S., Continuity of discrete homomorphisms of diffeomorphism groups, Geom. Topol., 19, 4, 2117-2154 (2015) · Zbl 1322.57026 [36] Jorquera, E., A universal nilpotent group of $$C^1$$ diffeomorphisms of the interval, Topol. Appl., 159, 8, 2115-2126 (2012) · Zbl 1267.57039 [37] Jorquera, E.; Navas, A.; Rivas, C., On the sharp regularity for arbitrary actions of nilpotent groups on the interval: the case of $$N_4$$, Ergod. Theory Dyn. Syst., 38, 1, 180-194 (2018) · Zbl 1387.37029 [38] Juschenko, K.; Monod, N., Cantor systems, piecewise translations and simple amenable groups, Ann. Math. (2), 178, 2, 775-787 (2013) · Zbl 1283.37011 [39] Kim, S.; Koberda, T., Anti-trees and right-angled Artin subgroups of braid groups, Geom. Topol., 19, 6, 3289-3306 (2015) · Zbl 1351.20021 [40] Kim, S.; Koberda, T., Free products and the algebraic structure of diffeomorphism groups, J. Topol., 11, 4, 1054-1076 (2018) · Zbl 1408.57020 [41] Kim, S.; Koberda, T.; Lodha, Y., Chain groups of homeomorphisms of the interval, Ann. Sci. Éc. Norm. Supér. (4), 52, 4, 797-820 (2019) · Zbl 07144472 [42] Koberda, T., Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups, Geom. Funct. Anal., 22, 6, 1541-1590 (2012) · Zbl 1282.37024 [43] Koberda, T., Lodha, Y.: Two-chains and square roots of Thompson’s group $$F$$. Ergod. Theory Dyn. Syst. (to appear) [44] Kopell, N.: Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). Am. Math. Soc., Provid., R.I., 1970, pp. 165-184 [45] Ling, W., Factorizable groups of homeomorphisms, Compos. Math., 51, 1, 41-50 (1984) · Zbl 0529.58009 [46] Lodha, Y.; Moore, JT, A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn., 10, 1, 177-200 (2016) · Zbl 1336.43001 [47] Mann, K., Homomorphisms between diffeomorphism groups, Ergod. Theory Dyn. Syst., 35, 1, 192-214 (2015) · Zbl 1311.57047 [48] Mann, K., Automatic continuity for homeomorphism groups and applications, Geom. Topol., 20, 5, 3033-3056 (2016) · Zbl 1362.57044 [49] Mann, K., A short proof that $${{\rm Diff}}_c(M)$$ is perfect, N. Y. J. Math., 22, 49-55 (2016) · Zbl 1335.57044 [50] Margulis, GA, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (1991), Berlin: Springer, Berlin [51] Mather, JN, Commutators of diffeomorphisms, Comment. Math. Helv., 49, 512-528 (1974) · Zbl 0289.57014 [52] Mather, JN, Commutators of diffeomorphisms. II, Comment. Math. Helv., 50, 33-40 (1975) · Zbl 0299.58008 [53] Moser, L., On the series, $$\sum 1/p$$, Am. Math. Mon., 65, 104-105 (1958) · Zbl 0081.27205 [54] Muller, M.P.: Sur l’approximation et l’instabilité des feuilletages (unpublished) [55] Navas, A., Actions de groupes de Kazhdan sur le cercle, Ann. Sci. École Norm. Sup. (4), 35, 5, 749-758 (2002) · Zbl 1028.58010 [56] Navas, A., Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal., 18, 3, 988-1028 (2008) · Zbl 1201.37060 [57] Navas, A., A finitely generated, locally indicable group with no faithful action by $$C^1$$ diffeomorphisms of the interval, Geom. Topol., 14, 1, 573-584 (2010) · Zbl 1197.37022 [58] Navas, A.: Groups of circle diffeomorphisms, Spanish ed., Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (2011) · Zbl 1236.37002 [59] Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math., 50, 1, 189-358 (1927) · JFM 53.0545.12 [60] Palis, J., Smale, S.: Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). Am. Math. Soc., Providence, R.I., 1970, pp. 223-231 [61] Parwani, K., $$C^1$$ actions on the mapping class groups on the circle, Algebr. Geom. Topol., 8, 2, 935-944 (2008) · Zbl 1155.37028 [62] Plante, JF; Thurston, WP, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv., 51, 4, 567-584 (1976) · Zbl 0348.57009 [63] Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68 · Zbl 0254.22005 [64] Šalát, T., On subseries, Math. Z., 85, 209-225 (1964) · Zbl 0127.28801 [65] Thurston, WP, Foliations and groups of diffeomorphisms, Bull. Am. Math. Soc., 80, 304-307 (1974) · Zbl 0295.57014 [66] Thurston, WP, A generalization of the Reeb stability theorem, Topology, 13, 347-352 (1974) · Zbl 0305.57025 [67] Thurston, WP, Existence of codimension-one foliations, Ann. Math. (2), 104, 2, 249-268 (1976) · Zbl 0347.57014 [68] Tsuboi, T.: $$\Gamma_1$$-structures avec une seule feuille, Astérisque (1984), no. 116, 222-234, Transversal structure of foliations (Toulouse, 1982) [69] Tsuboi, T., Examples of nonsmoothable actions on the interval, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34, 2, 271-274 (1987) · Zbl 0671.58018 [70] Dave Witte Morris, Arithmetic groups of higher $${ Q}$$-rank cannot act on $$1$$-manifolds, Proc. Am. Math. Soc., 122, 2, 333-340 (1994) · Zbl 0818.22006
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