2018, 13: 147-161. doi: 10.3934/jmd.2018015

Distortion and the automorphism group of a shift

1. 

Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA

2. 

Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

3. 

Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France

Dedicated to the memory of Roy Adler

Received  December 15, 2016 Revised  February 22, 2017 Published  December 2018

Fund Project: BK: Partially supported by NSF grant 1500670

The set of automorphisms of a one-dimensional subshift $(X, σ)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy subshift. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm top}}(X) = 0$. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

Citation: Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015
References:
[1]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614. doi: 10.1112/plms/s3-25.4.603.

[2]

M. BoyleD. 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]

V. Cyr, J. Franks and B. Kra, The spacetime of a shift automorphism, Trans. Amer. Math. Soc., 371 (2019), 461–488. doi: 10.1090/tran/7254.

[4]

V. Cyr and B. Kra, Nonexpansive $ \mathbb{Z}^2$-subdynamics and Nivat's conjecture, Trans. Amer. Math. Soc., 367 (2015), 6487-6537. doi: 10.1090/S0002-9947-2015-06391-0.

[5]

V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp. doi: 10.1017/fms.2015.3.

[6]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495. doi: 10.3934/jmd.2016.10.483.

[7]

S. DonosoF. DurandA. 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]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.

[9]

Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Ser. A-B, 272 (1971), A1695-A1696.

[10]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory., 3 (1969), 320-375. doi: 10.1007/BF01691062.

[11]

M. Hochman, Non-expansive directions for $ \mathbb{Z}^2$ actions, Ergodic Theory Dynam. Systems., 31 (2011), 91-112. doi: 10.1017/S0143385709001084.

[12]

H. Keynes and J. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59. doi: 10.1007/BF01695625.

[13]

K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.

[14]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.

[15]

A. LubotzkyS. Mozes and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math., 91 (2000), 5-53.

[16]

A. I. Mal'cev, Generalized nilpotent algebras and their associated groups, (Russian) Mat. Sbornik N.S., 25 (1949), 347-366.

[17]

A. I. Mal'cev, Nilpotent torsion-free groups, (Russian) Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 201-212.

[18]

A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, 395, Cambridge University Press, Cambridge, 2012.

[19]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6.

[20]

S. Meskin, Nonresidually finite one-relator groups, Tran. Amer. Math. Soc., 164 (1972), 105-114. doi: 10.1090/S0002-9947-1972-0285589-5.

[21]

M. Morse and G. A. Hedlund, Symbolic Dynamics, Amer. J. Math., 60 (1938), 815-866. doi: 10.2307/2371264.

[22]

M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[23]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, 1972.

[24]

H. V. Waldinger and A. M. Gaglione, On nilpotent products of cyclic groups reexamined by the commutator calculus, Can. J. Math., 27 (1975), 1185-1210. doi: 10.4153/CJM-1975-125-9.

show all references

References:
[1]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614. doi: 10.1112/plms/s3-25.4.603.

[2]

M. BoyleD. 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]

V. Cyr, J. Franks and B. Kra, The spacetime of a shift automorphism, Trans. Amer. Math. Soc., 371 (2019), 461–488. doi: 10.1090/tran/7254.

[4]

V. Cyr and B. Kra, Nonexpansive $ \mathbb{Z}^2$-subdynamics and Nivat's conjecture, Trans. Amer. Math. Soc., 367 (2015), 6487-6537. doi: 10.1090/S0002-9947-2015-06391-0.

[5]

V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp. doi: 10.1017/fms.2015.3.

[6]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495. doi: 10.3934/jmd.2016.10.483.

[7]

S. DonosoF. DurandA. 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]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.

[9]

Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Ser. A-B, 272 (1971), A1695-A1696.

[10]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory., 3 (1969), 320-375. doi: 10.1007/BF01691062.

[11]

M. Hochman, Non-expansive directions for $ \mathbb{Z}^2$ actions, Ergodic Theory Dynam. Systems., 31 (2011), 91-112. doi: 10.1017/S0143385709001084.

[12]

H. Keynes and J. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59. doi: 10.1007/BF01695625.

[13]

K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.

[14]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.

[15]

A. LubotzkyS. Mozes and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math., 91 (2000), 5-53.

[16]

A. I. Mal'cev, Generalized nilpotent algebras and their associated groups, (Russian) Mat. Sbornik N.S., 25 (1949), 347-366.

[17]

A. I. Mal'cev, Nilpotent torsion-free groups, (Russian) Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 201-212.

[18]

A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, 395, Cambridge University Press, Cambridge, 2012.

[19]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6.

[20]

S. Meskin, Nonresidually finite one-relator groups, Tran. Amer. Math. Soc., 164 (1972), 105-114. doi: 10.1090/S0002-9947-1972-0285589-5.

[21]

M. Morse and G. A. Hedlund, Symbolic Dynamics, Amer. J. Math., 60 (1938), 815-866. doi: 10.2307/2371264.

[22]

M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.

[23]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, 1972.

[24]

H. V. Waldinger and A. M. Gaglione, On nilpotent products of cyclic groups reexamined by the commutator calculus, Can. J. Math., 27 (1975), 1185-1210. doi: 10.4153/CJM-1975-125-9.

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