\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Direct products, overlapping actions, and critical regularity

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $ H $ and $ K $ are two non-solvable groups then a faithful $ C^{1,\tau} $ action of $ H\times K $ on a compact interval $ I $ is not overlapping for all $ \tau>0 $, which by definition means that there must be non-trivial $ h\in H $ and $ k\in K $ with disjoint support. As a corollary we prove that the right-angled Artin group $ (F_2\times F_2)*\mathbb{Z} $ has critical regularity one, which is to say that it admits a faithful $ C^1 $ action on $ I $, but no faithful $ C^{1,\tau} $ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group $ F $ does not admit a faithful $ C^1 $ overlapping action on $ I $, so that $ F*\mathbb{Z} $ is a new example of a locally indicable group admitting no faithful $ C^1 $ action on $ I $.

    Mathematics Subject Classification: Primary: 57M60; Secondary: 20F36, 37C05, 37C85, 20F14, 20F60.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The two-chain criterion

  • [1] H. BaikS. Kim and T. Koberda, Right-angled Artin groups in the $C^\infty$ diffeomorphism group of the real line, Israel J. Math., 213 (2016), 175-182.  doi: 10.1007/s11856-016-1307-8.
    [2] H. BaikS. Kim and T. Koberda, Unsmoothable group actions on compact one-manifolds, J. Eur. Math. Soc. (JEMS), 21 (2019), 2333-2353.  doi: 10.4171/JEMS/886.
    [3] C. Bonatti and É. Farinelli, Centralizers of $C^1$-contractions of the half line, Groups Geom. Dyn., 9 (2015), 831-889.  doi: 10.4171/GGD/330.
    [4] C. BonattiI. MonteverdeA. Navas and C. Rivas, Rigidity for $C^1$ actions on the interval arising from hyperbolicity I: Solvable groups, Math. Z., 286 (2017), 919-949.  doi: 10.1007/s00209-016-1790-y.
    [5] J. Brum, N. Matte Bon, C. Rivas and M. Triestino, Locally moving groups acting on the line and $ \mathbb{R}$-focal actions, preprint, arXiv: 2104.14678.
    [6] D. Calegari, Nonsmoothable, locally indicable group actions on the interval, Algebr. Geom. Topol., 8 (2008), 609-613.  doi: 10.2140/agt.2008.8.609.
    [7] G. CastroE. Jorquera and A. Navas, Sharp regularity for certain nilpotent group actions on the interval, Math. Ann., 359 (2014), 101-152.  doi: 10.1007/s00208-013-0995-1.
    [8] B. DeroinV. Kleptsyn and A. Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007), 199-262.  doi: 10.1007/s11511-007-0020-1.
    [9] G. Duchamp and D. Krob, The lower central series of the free partially commutative group, Semigroup Forum, 45 (1992), 385-394.  doi: 10.1007/BF03025778.
    [10] B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds. III. {N}ilpotent subgroups, Ergodic Theory Dynam. Systems, 23 (2003), 1467-1484.  doi: 10.1017/S0143385702001712.
    [11] É. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62 (1987), 185-239.  doi: 10.1007/BF02564445.
    [12] R. I. Grigorchuk and A. Machì, On a group of intermediate growth that acts on a line by homeomorphisms, Mathematical Notes, 53 (1993), 146-157.  doi: 10.1007/BF01208318.
    [13] E. Jorquera, A universal nilpotent group of $C^1$ diffeomorphisms of the interval, Topology Appl., 159 (2012), 2115-2126.  doi: 10.1016/j.topol.2012.02.003.
    [14] E. JorqueraA. Navas and C. Rivas, On the sharp regularity for arbitrary actions of nilpotent groups on the interval: The case of $N_4$, Ergodic Theory Dynam. Systems, 38 (2018), 180-194.  doi: 10.1017/etds.2016.38.
    [15] M. Kambites, On commuting elements and embeddings of graph groups and monoids, Proc. Edinb. Math. Soc. (2), 52 (2009), 155-170.  doi: 10.1017/S0013091507000119.
    [16] S. Kim and T. Koberda, Embedability between right-angled {A}rtin groups, Geom. Topol., 17 (2013), 493-530.  doi: 10.2140/gt.2013.17.493.
    [17] S. Kim and T. Koberda, Free products and the algebraic structure of diffeomorphism groups, J. Topol., 11 (2018), 1054-1076.  doi: 10.1112/topo.12079.
    [18] S. Kim and T. Koberda, Diffeomorphism groups of critical regularity, Invent. Math., 221 (2020), 421-501.  doi: 10.1007/s00222-020-00953-y.
    [19] S. Kim, T. Koberda and Y. Lodha, Chain groups of homeomorphisms of the interval, Ann. Sci. Éc. Norm. Supér. (4), 52 (2019), 797–820. doi: 10.24033/asens.2397.
    [20] M. P. Muller, Sur l'approximation et l'instabilité des feuilletages, unpublished.
    [21] A. Navas, Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal., 18 (2008), 988-1028.  doi: 10.1007/s00039-008-0667-6.
    [22] A. Navas, A finitely generated, locally indicable group with no faithful action by $C^1$ diffeomorphisms of the interval, Geom. Topol., 14 (2010), 573-584.  doi: 10.2140/gt.2010.14.573.
    [23] A. Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble), 60 (2010), 1685-1740.  doi: 10.5802/aif.2570.
    [24] A. NavasGroups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011.  doi: 10.7208/chicago/9780226569505.001.0001.
    [25] A. Navas, An example concerning the theory of levels for codimension-one foliations, Publ. Mat. Urug., 12 (2011), 169-176. 
    [26] A. Navas and C. Rivas, A new characterization of {C}onrad's property for group orderings, with applications, Algebr. Geom. Top., 9 (2009), 2079-2100.  doi: 10.2140/agt.2009.9.2079.
    [27] J.-P. Serre, Arbres, Amalgames, $ \mathrm{SL}_{2}$, Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, 46, Société Mathématique de France, Paris, 1977.
    [28] W. P. Thurston, A generalization of the {R}eeb stability theorem, Topology, 13 (1974), 347-352.  doi: 10.1016/0040-9383(74)90025-1.
    [29] T. Tsuboi, Foliated cobordism classes of certain foliated $S^{1}$-bundles over surfaces, Topology, 23 (1984), 233-244.  doi: 10.1016/0040-9383(84)90042-9.
    [30] T. Tsuboi, Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, J. Math. Soc. Japan, 47 (1995), 1-30.  doi: 10.2969/jmsj/04710001.
  • 加载中

Figures(1)

SHARE

Article Metrics

HTML views(1144) PDF downloads(133) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return