Topological full groups of minimal subshifts with subgroups of intermediate growth

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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

Abstract
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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.
[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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