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

Topological full groups of minimal subshifts with subgroups of intermediate growth

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”.
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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

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