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2015, 9: 67-80. doi: 10.3934/jmd.2015.9.67

Topological full groups of minimal subshifts with subgroups of intermediate growth

Nicolás Matte Bon 1,

1. 

Laboratoire de Mathémathiques d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex & DMA, École Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France

Received  August 2014 Revised  January 2015 Published  May 2015


This work is partially supported by the ERC starting grant GA 257110 “RaWG”. We show that every Grigorchuk group $G_\omega$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; moreover it can have finitely generated infinite torsion subgroups, answering a question of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_\omega$ can be embedded in a finitely generated simple group that has trivial Poisson boundary for every simple random walk.

    This work is partially supported by the ERC starting grant GA 257110 “RaWG”.
Keywords: Cantor minimal systems, growth of groups, periodic groups..
Mathematics Subject Classification: Primary: 37B10; Secondary: 20F69, 20F6.
Citation: Nicolás Matte Bon. Topological full groups of minimal subshifts with subgroups of intermediate growth. Journal of Modern Dynamics, 2015, 9: 67-80. doi: 10.3934/jmd.2015.9.67
References:
[1]

A. Avez, Théorème de Choquet-Deny pour les groupes à croissance non exponentielle, C. R. Acad. Sci. Paris Sér. A, 279 (1974), 25-28.  Google Scholar

[2]

L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova (Din. Sist., Avtom. i Beskon. Gruppy), 231 (2000), 5-45. Google Scholar

[3]

L. Bartholdi, R. I. Grigorchuk and Z. Šuniḱ, Branch groups, in Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003, 989-1112. doi: 10.1016/S1570-7954(03)80078-5.  Google Scholar

[4]

J. Cassaigne and F. Nicolas, Factor complexity, in Combinatorics, Automata and Number Theory, Encyclopedia Math. Appl., 135, Cambridge Univ. Press, Cambridge, 2010, 163-247.  Google Scholar

[5]

Y. Cornulier, Groupes pleins-topologiques [d'après Matui, Juschenko, Monod,...], Astérisque, Séminaire Bourbaki, Vol. 2012/2013, (361), Exp. No. 1064, 2014. Google Scholar

[6]

G. Elek and N. Monod, On the topological full group of a minimal Cantor $\mathbbZ^2$-system, Proc. Amer. Math. Soc., 141 (2013), 3549-3552. doi: 10.1090/S0002-9939-2013-11654-0.  Google Scholar

[7]

R. Grigorchuk, D. Lenz, and T. Nagnibeda, Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order, preprint, arXiv:1412.6822, 2014. Google Scholar

[8]

A. P. Gorjuškin, Imbedding of countable groups in $2$-generator simple groups, Mat. Zametki, 16 (1974), 231-235.  Google Scholar

[9]

W. H. Gottschalk, Almost period points with respect to transformation semi-groups, Ann. of Math. (2), 47 (1946), 762-766. doi: 10.2307/1969233.  Google Scholar

[10]

T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689.  Google Scholar

[11]

R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985.  Google Scholar

[12]

P. Hall, On the embedding of a group in a join of given groups, Collection of articles dedicated to the memory of Hanna Neumann, VIII, J. Austral. Math. Soc., 17 (1974), 434-495. doi: 10.1017/S1446788700018073.  Google Scholar

[13]

K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787. doi: 10.4007/annals.2013.178.2.7.  Google Scholar

[14]

V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Probab., 11 (1983), 457-490. doi: 10.1214/aop/1176993497.  Google Scholar

[15]

H. Matui, Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251. doi: 10.1142/S0129167X06003448.  Google Scholar

[16]

H. Matui, Some remarks on topological full groups of Cantor minimal systems II, Ergodic Theory Dynam. Systems, 33 (2013), 1542-1549. doi: 10.1017/S0143385712000399.  Google Scholar

[17]

N. Matte Bon, Subshifts with slow complexity and simple groups with the Liouville property, Geom. Funct. Anal., 24 (2014), 1637-1659. doi: 10.1007/s00039-014-0293-4.  Google Scholar

[18]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.  Google Scholar

[19]

P. E. Schupp, Embeddings into simple groups, J. London Math. Soc. (2), 13 (1976), 90-94.  Google Scholar

[20]

E. K. van Douwen, Measures invariant under actions of $F_2$, Topology Appl., 34 (1990), 53-68. doi: 10.1016/0166-8641(90)90089-K.  Google Scholar

[21]

Ya. Vorobets, On a substitution subshift related to the Grigorchuk group, Tr. Mat. Inst. Steklova, (Differentsialnye Uravneniya i Topologiya. II), 271 (2010), 319-334. doi: 10.1134/S0081543810040218.  Google Scholar

[22]

Ya. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 221-248. doi: 10.1090/conm/567/11250.  Google Scholar

show all references

References:
[1]

A. Avez, Théorème de Choquet-Deny pour les groupes à croissance non exponentielle, C. R. Acad. Sci. Paris Sér. A, 279 (1974), 25-28.  Google Scholar

[2]

L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova (Din. Sist., Avtom. i Beskon. Gruppy), 231 (2000), 5-45. Google Scholar

[3]

L. Bartholdi, R. I. Grigorchuk and Z. Šuniḱ, Branch groups, in Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003, 989-1112. doi: 10.1016/S1570-7954(03)80078-5.  Google Scholar

[4]

J. Cassaigne and F. Nicolas, Factor complexity, in Combinatorics, Automata and Number Theory, Encyclopedia Math. Appl., 135, Cambridge Univ. Press, Cambridge, 2010, 163-247.  Google Scholar

[5]

Y. Cornulier, Groupes pleins-topologiques [d'après Matui, Juschenko, Monod,...], Astérisque, Séminaire Bourbaki, Vol. 2012/2013, (361), Exp. No. 1064, 2014. Google Scholar

[6]

G. Elek and N. Monod, On the topological full group of a minimal Cantor $\mathbbZ^2$-system, Proc. Amer. Math. Soc., 141 (2013), 3549-3552. doi: 10.1090/S0002-9939-2013-11654-0.  Google Scholar

[7]

R. Grigorchuk, D. Lenz, and T. Nagnibeda, Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order, preprint, arXiv:1412.6822, 2014. Google Scholar

[8]

A. P. Gorjuškin, Imbedding of countable groups in $2$-generator simple groups, Mat. Zametki, 16 (1974), 231-235.  Google Scholar

[9]

W. H. Gottschalk, Almost period points with respect to transformation semi-groups, Ann. of Math. (2), 47 (1946), 762-766. doi: 10.2307/1969233.  Google Scholar

[10]

T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689.  Google Scholar

[11]

R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985.  Google Scholar

[12]

P. Hall, On the embedding of a group in a join of given groups, Collection of articles dedicated to the memory of Hanna Neumann, VIII, J. Austral. Math. Soc., 17 (1974), 434-495. doi: 10.1017/S1446788700018073.  Google Scholar

[13]

K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787. doi: 10.4007/annals.2013.178.2.7.  Google Scholar

[14]

V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Probab., 11 (1983), 457-490. doi: 10.1214/aop/1176993497.  Google Scholar

[15]

H. Matui, Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251. doi: 10.1142/S0129167X06003448.  Google Scholar

[16]

H. Matui, Some remarks on topological full groups of Cantor minimal systems II, Ergodic Theory Dynam. Systems, 33 (2013), 1542-1549. doi: 10.1017/S0143385712000399.  Google Scholar

[17]

N. Matte Bon, Subshifts with slow complexity and simple groups with the Liouville property, Geom. Funct. Anal., 24 (2014), 1637-1659. doi: 10.1007/s00039-014-0293-4.  Google Scholar

[18]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.  Google Scholar

[19]

P. E. Schupp, Embeddings into simple groups, J. London Math. Soc. (2), 13 (1976), 90-94.  Google Scholar

[20]

E. K. van Douwen, Measures invariant under actions of $F_2$, Topology Appl., 34 (1990), 53-68. doi: 10.1016/0166-8641(90)90089-K.  Google Scholar

[21]

Ya. Vorobets, On a substitution subshift related to the Grigorchuk group, Tr. Mat. Inst. Steklova, (Differentsialnye Uravneniya i Topologiya. II), 271 (2010), 319-334. doi: 10.1134/S0081543810040218.  Google Scholar

[22]

Ya. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 221-248. doi: 10.1090/conm/567/11250.  Google Scholar

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