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Spectral killers and Poisson bracket invariants
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 |
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”.
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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