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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.   Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.   Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (73)
  • HTML views (515)
  • Cited by (0)

Other articles
by authors

[Back to Top]