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Normal amenable subgroups of the automorphism group of the full shift



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.



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[1] Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71114.
[2] Coven, E., Quas, A. and Yassawi, R.. Automorphisms of some Toeplitz and other minimal shifts with sublinear complexity. Preprint, 2015, arXiv:1505.02482.
[3] Cyr, V. and Kra, B.. The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc. 144 (2016), 613621.
[4] Cyr, V. and Kra, B.. The automorphism group of a minimal shift of stretched exponential growth. J. Mod. Dynam. 10 (2016), 483495.
[5] Cyr, V. and Kra, B.. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma 3 (2015), e5.
[6] Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity minimal subshifts. Preprint, 2015, arXiv:1501.00510.
[7] Furman, A.. On minimal, strongly proximal actions of locally compact groups. Israel J. Math. 136(1) (2003), 173187.
[8] Furstenberg, H.. A Poisson formula for semi-simple Lie groups. Ann. of Math. (2) 77 (1963), 335386.
[9] Glasner, S.. Topological dynamics and group theory. Trans. Amer. Math. Soc. 187 (1974), 327334.
[10] Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3(4) (1969), 320375.
[11] Hochman, M.. On the automorphism groups of multidimensional shifts of finite type. Ergod. Th. & Dynam. Sys. 30(3) (2010), 809840.
[12] Nevo, A.. Boundary theory and harmonic analysis on boundary transitive graphs. Amer. J. Math. 116(2) (1994), 243282.
[13] Ryan, J. P.. The shift and commutativity. Math. Systems Theory 6 (1972), 8285.
[14] Salo, V.. Toeplitz subshift whose automorphism group is not finitely generated. Preprint, 2014, arXiv:1411.3299.
[15] Salo, V. and Törmä, I.. Block maps between primitive uniform and Pisot substitutions. Ergod. Th. & Dynam. Sys. 35(7) (2014), 119.


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