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Normal amenable subgroups of the automorphism group of the full shift

Published online by Cambridge University Press:  07 September 2017

JOSHUA FRISCH ,
TOMER SCHLANK  and
OMER TAMUZ
JOSHUA FRISCH
Affiliation:
California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA email [email protected], [email protected]
TOMER SCHLANK
Affiliation:
Hebrew University, Givat Ram, Jerusalem, Israel email [email protected]
OMER TAMUZ
Affiliation:
California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA email [email protected], [email protected]
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Abstract

We show that every normal amenable subgroup of the automorphism group of the full shift is contained in its center. This follows from the analysis of this group’s Furstenberg topological boundary, through the construction of a minimal and strongly proximal action. We extend this result to higher dimensional full shifts. This also provides a new proof of Ryan’s theorem and of the fact that these groups contain free groups.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1290 - 1298
Copyright
© Cambridge University Press, 2017 

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References

Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71–114.Google Scholar
Coven, E., Quas, A. and Yassawi, R.. Automorphisms of some Toeplitz and other minimal shifts with sublinear complexity. Preprint, 2015, arXiv:1505.02482.Google Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc. 144 (2016), 613–621.Google Scholar
Cyr, V. and Kra, B.. The automorphism group of a minimal shift of stretched exponential growth. J. Mod. Dynam. 10 (2016), 483–495.Google Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma 3 (2015), e5.Google Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity minimal subshifts. Preprint, 2015, arXiv:1501.00510.Google Scholar
Furman, A.. On minimal, strongly proximal actions of locally compact groups. Israel J. Math. 136(1) (2003), 173–187.Google Scholar
Furstenberg, H.. A Poisson formula for semi-simple Lie groups. Ann. of Math. (2) 77 (1963), 335–386.Google Scholar
Glasner, S.. Topological dynamics and group theory. Trans. Amer. Math. Soc. 187 (1974), 327–334.Google Scholar
Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3(4) (1969), 320–375.Google Scholar
Hochman, M.. On the automorphism groups of multidimensional shifts of finite type. Ergod. Th. & Dynam. Sys. 30(3) (2010), 809–840.Google Scholar
Nevo, A.. Boundary theory and harmonic analysis on boundary transitive graphs. Amer. J. Math. 116(2) (1994), 243–282.Google Scholar
Ryan, J. P.. The shift and commutativity. Math. Systems Theory 6 (1972), 82–85.Google Scholar
Salo, V.. Toeplitz subshift whose automorphism group is not finitely generated. Preprint, 2014, arXiv:1411.3299.Google Scholar
Salo, V. and Törmä, I.. Block maps between primitive uniform and Pisot substitutions. Ergod. Th. & Dynam. Sys. 35(7) (2014), 1–19.Google Scholar