# American Institute of Mathematical Sciences

2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483

## The automorphism group of a minimal shift of stretched exponential growth

 1 Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837, United States 2 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, United States

Received  September 2015 Revised  August 2016 Published  October 2016

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent.
Citation: Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483
##### References:
 [1] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614. [2] M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2. [3] E. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts, using atypical equivalence classes, Discrete Anal., (2016), 1-28. doi: 10.19086/da.611. [4] V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621. doi: 10.1090/proc12719. [5] V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27pp. doi: 10.1017/fms.2015.3. [6] P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. [7] S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95. doi: 10.1017/etds.2015.70. [8] S. Donoso, F. Durand, A. Maass and S. Petite, Private communication. [9] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73. [10] Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696. [11] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062. [12] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431. [13] K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230. [14] V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016. [15] V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory and Dynam. Systems, 35 (2015), 2292-2310. doi: 10.1017/etds.2014.29. [16] L. van den Dries and A. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra, 89 (1984), 349-374. doi: 10.1016/0021-8693(84)90223-0.

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##### References:
 [1] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614. [2] M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2. [3] E. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts, using atypical equivalence classes, Discrete Anal., (2016), 1-28. doi: 10.19086/da.611. [4] V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621. doi: 10.1090/proc12719. [5] V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27pp. doi: 10.1017/fms.2015.3. [6] P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. [7] S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95. doi: 10.1017/etds.2015.70. [8] S. Donoso, F. Durand, A. Maass and S. Petite, Private communication. [9] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73. [10] Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696. [11] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062. [12] M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431. [13] K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230. [14] V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016. [15] V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory and Dynam. Systems, 35 (2015), 2292-2310. doi: 10.1017/etds.2014.29. [16] L. van den Dries and A. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra, 89 (1984), 349-374. doi: 10.1016/0021-8693(84)90223-0.
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