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2021, 17: 285-304.
doi: 10.3934/jmd.2021009
Direct products, overlapping actions, and critical regularity
| 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 |
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
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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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. Navas, Groups 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. |
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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. Navas, Groups 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. |
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