American Institute of Mathematical Sciences

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  2021, 17: 285-304. doi: 10.3934/jmd.2021009

Direct products, overlapping actions, and critical regularity

Sang-hyun Kim 1, , Thomas Koberda 2, and  Cristóbal Rivas 3,

1. 

School of Mathematics, Korea Institute for Advanced Study (KIAS), Seoul, 02455, Korea
2. 

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
3. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Alameda 3363, Santiago, Chile

Received  October 14, 2020 Revised  May 05, 2021 Published  June 2021

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 $.

Keywords: Free product, non-solvable group, Thompson's group, right-angled Artin group, smoothing, critical regularity, lamplighter group.
Mathematics Subject Classification: Primary: 57M60; Secondary: 20F36, 37C05, 37C85, 20F14, 20F60.
Citation: Sang-hyun Kim, Thomas Koberda, Cristóbal Rivas. Direct products, overlapping actions, and critical regularity. Journal of Modern Dynamics, 2021, 17: 285-304. doi: 10.3934/jmd.2021009
References:
[1]

H. Baik, S. 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.  Google Scholar

[2]

H. Baik, S. 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.  Google Scholar

[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.  Google Scholar

[4]

C. Bonatti, I. Monteverde, A. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

G. Castro, E. 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.  Google Scholar

[8]

B. Deroin, V. Kleptsyn and A. Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007), 199-262.  doi: 10.1007/s11511-007-0020-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Jorquera, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

S. Kim and T. Koberda, Diffeomorphism groups of critical regularity, Invent. Math., 221 (2020), 421-501.  doi: 10.1007/s00222-020-00953-y.  Google Scholar

[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.  Google Scholar

[20]

M. P. Muller, Sur l'approximation et l'instabilité des feuilletages, unpublished. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble), 60 (2010), 1685-1740.  doi: 10.5802/aif.2570.  Google Scholar

[24] A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011.  doi: 10.7208/chicago/9780226569505.001.0001.  Google Scholar
[25]

A. Navas, An example concerning the theory of levels for codimension-one foliations, Publ. Mat. Urug., 12 (2011), 169-176.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Baik, S. 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.  Google Scholar

[2]

H. Baik, S. 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.  Google Scholar

[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.  Google Scholar

[4]

C. Bonatti, I. Monteverde, A. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

G. Castro, E. 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.  Google Scholar

[8]

B. Deroin, V. Kleptsyn and A. Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007), 199-262.  doi: 10.1007/s11511-007-0020-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Jorquera, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

S. Kim and T. Koberda, Diffeomorphism groups of critical regularity, Invent. Math., 221 (2020), 421-501.  doi: 10.1007/s00222-020-00953-y.  Google Scholar

[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.  Google Scholar

[20]

M. P. Muller, Sur l'approximation et l'instabilité des feuilletages, unpublished. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble), 60 (2010), 1685-1740.  doi: 10.5802/aif.2570.  Google Scholar

[24] A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011.  doi: 10.7208/chicago/9780226569505.001.0001.  Google Scholar
[25]

A. Navas, An example concerning the theory of levels for codimension-one foliations, Publ. Mat. Urug., 12 (2011), 169-176.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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