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Realization of big centralizers of minimal aperiodic actions on the Cantor set
1. | Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Avenida Libertador Bernardo O'Higgins 3363, Santiago, Chile |
2. | Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS-UMR 7352, Université de Picardie Jules Verne, 33, rue Saint Leu 80039 Amiens Cedex 1, France |
In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $ \mathbb Z $ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $ G $, the centralizer of any minimal aperiodic $ G $-action on the Cantor set is a subgroup of the centralizer of a minimal $ \mathbb Z $-action.
References:
[1] |
N. Aubrun, S. Barbieri and S. Thomassé,
Realization of aperiodic subshifts and uniform densities in groups, Groups Geom. Dyn., 13 (2019), 107-129.
doi: 10.4171/GGD/487. |
[2] |
M. Baake, J. A. G. Roberts and R. Yassawi,
Reversing and extended symmetries of shift spaces, Discrete and Continuous Dynamical Systems, 38 (2018), 835-866.
doi: 10.3934/dcds.2018036. |
[3] |
M. Boyle, D. Lind and D. Rudolph,
The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114.
doi: 10.1090/S0002-9947-1988-0927684-2. |
[4] |
W. Bulatek and J. Kwiatkowski,
Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers, Studia Math., 103 (1992), 133-142.
doi: 10.4064/sm-103-2-133-142. |
[5] |
T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Cellular automata, Encycl. Complex. Syst. Sci., Springer, New York, (2018), 221–238. |
[6] |
M. I. Cortez and S. Petite,
$G$-odometers and their almost one-to-one extensions, J. London Math. Soc. (2), 78 (2008), 1-20.
doi: 10.1112/jlms/jdn002. |
[7] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), 28 pp.
doi: 10.19086/da.611. |
[8] |
V. Cyr and B. Kra,
The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621.
doi: 10.1090/proc12719. |
[9] |
V. Cyr and B. Kra, The automorphism group of a shift of linear growth: Beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp.
doi: 10.1017/fms.2015.3. |
[10] |
V. Cyr and B. Kra,
The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.
doi: 10.3934/jmd.2016.10.483. |
[11] |
V. Cyr, J. Franks, B. Kra and S. Petite,
Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147-161.
doi: 10.3934/jmd.2018015. |
[12] |
S. Donoso, F. Durand, A. Maass and S. Petite,
On automorphism groups of low complexity subshifts, Ergodic Theory and Dynam. Systems, 36 (2016), 64-95.
doi: 10.1017/etds.2015.70. |
[13] |
S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal., 2017 (2017), 19 pp. |
[14] |
F. Durand, N. Ormes and S. Petite,
Self-induced systems, J. Anal. Math., 135 (2018), 725-756.
doi: 10.1007/s11854-018-0051-x. |
[15] |
T. Giordano, I. F. Putnam and C. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[16] |
E. Glasner, T. Tsankov, B. Weiss and A. Zucker, Bernoulli disjointness, Preprint, arXiv: 1901.03406. |
[17] |
G. Hjorth and M. Molberg,
Free continuous actions on zero-dimensional spaces, Topology Appl., 153 (2006), 1116-1131.
doi: 10.1016/j.topol.2005.03.003. |
[18] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations and simple amenable groups, Ann. of Math., 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[19] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[20] |
H. Matui,
Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251.
doi: 10.1142/S0129167X06003448. |
[21] |
K. Medynets,
Reconstruction of orbits of Cantor systems from full groups, Bull. Lond. Math. Soc., 43 (2011), 1104-1110.
doi: 10.1112/blms/bdr045. |
[22] |
K. Medynets and J. P. Talisse,
Toeplitz subshifts with trivial centralizers and positive entropy, Involve, 12 (2019), 395-410.
doi: 10.2140/involve.2019.12.395. |
[23] |
V. Salo and M. Schraudner, Automorphism groups of subshifts via group extensions, preprint. |
[24] |
D. Witte,
Arithmetic groups of higher Q-rank cannot act on $1$-manifolds, Proc. Amer. Math. Soc., 122 (1994), 333-340.
doi: 10.2307/2161021. |
show all references
References:
[1] |
N. Aubrun, S. Barbieri and S. Thomassé,
Realization of aperiodic subshifts and uniform densities in groups, Groups Geom. Dyn., 13 (2019), 107-129.
doi: 10.4171/GGD/487. |
[2] |
M. Baake, J. A. G. Roberts and R. Yassawi,
Reversing and extended symmetries of shift spaces, Discrete and Continuous Dynamical Systems, 38 (2018), 835-866.
doi: 10.3934/dcds.2018036. |
[3] |
M. Boyle, D. Lind and D. Rudolph,
The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114.
doi: 10.1090/S0002-9947-1988-0927684-2. |
[4] |
W. Bulatek and J. Kwiatkowski,
Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers, Studia Math., 103 (1992), 133-142.
doi: 10.4064/sm-103-2-133-142. |
[5] |
T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Cellular automata, Encycl. Complex. Syst. Sci., Springer, New York, (2018), 221–238. |
[6] |
M. I. Cortez and S. Petite,
$G$-odometers and their almost one-to-one extensions, J. London Math. Soc. (2), 78 (2008), 1-20.
doi: 10.1112/jlms/jdn002. |
[7] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), 28 pp.
doi: 10.19086/da.611. |
[8] |
V. Cyr and B. Kra,
The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621.
doi: 10.1090/proc12719. |
[9] |
V. Cyr and B. Kra, The automorphism group of a shift of linear growth: Beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp.
doi: 10.1017/fms.2015.3. |
[10] |
V. Cyr and B. Kra,
The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.
doi: 10.3934/jmd.2016.10.483. |
[11] |
V. Cyr, J. Franks, B. Kra and S. Petite,
Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147-161.
doi: 10.3934/jmd.2018015. |
[12] |
S. Donoso, F. Durand, A. Maass and S. Petite,
On automorphism groups of low complexity subshifts, Ergodic Theory and Dynam. Systems, 36 (2016), 64-95.
doi: 10.1017/etds.2015.70. |
[13] |
S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal., 2017 (2017), 19 pp. |
[14] |
F. Durand, N. Ormes and S. Petite,
Self-induced systems, J. Anal. Math., 135 (2018), 725-756.
doi: 10.1007/s11854-018-0051-x. |
[15] |
T. Giordano, I. F. Putnam and C. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[16] |
E. Glasner, T. Tsankov, B. Weiss and A. Zucker, Bernoulli disjointness, Preprint, arXiv: 1901.03406. |
[17] |
G. Hjorth and M. Molberg,
Free continuous actions on zero-dimensional spaces, Topology Appl., 153 (2006), 1116-1131.
doi: 10.1016/j.topol.2005.03.003. |
[18] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations and simple amenable groups, Ann. of Math., 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[19] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[20] |
H. Matui,
Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251.
doi: 10.1142/S0129167X06003448. |
[21] |
K. Medynets,
Reconstruction of orbits of Cantor systems from full groups, Bull. Lond. Math. Soc., 43 (2011), 1104-1110.
doi: 10.1112/blms/bdr045. |
[22] |
K. Medynets and J. P. Talisse,
Toeplitz subshifts with trivial centralizers and positive entropy, Involve, 12 (2019), 395-410.
doi: 10.2140/involve.2019.12.395. |
[23] |
V. Salo and M. Schraudner, Automorphism groups of subshifts via group extensions, preprint. |
[24] |
D. Witte,
Arithmetic groups of higher Q-rank cannot act on $1$-manifolds, Proc. Amer. Math. Soc., 122 (1994), 333-340.
doi: 10.2307/2161021. |
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