July  2013, 7(3): 369-394. doi: 10.3934/jmd.2013.7.369

Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface

1. 

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States

2. 

Department of Mathematics and Computer Science, Lehman College, Bronx, NY 10468, United States

Received  December 2012 Revised  September 2013 Published  December 2013

We show that if $M$ is a compact oriented surface of genus $0$ and $G$ is a subgroup of Symp$^\omega_\mu(M)$ that has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in$ Symp$^\omega_\mu(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of Symp$^\omega_\mu(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Symp$^\omega_\mu(M)$.
Citation: John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369
References:
[1]

M. Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991.  Google Scholar

[2]

M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out($F^n$). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2), 151 (2000), 517-623. doi: 10.2307/121043.  Google Scholar

[3]

M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out($F^n$). II. A Kolchin type theorem, Ann. of Math. (2), 161 (2005), 1-59. doi: 10.4007/annals.2005.161.1.  Google Scholar

[4]

E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42.  Google Scholar

[5]

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc., 91 (1984), 503-504. doi: 10.2307/2045329.  Google Scholar

[6]

B. Farb and P. Shalen, Groups of real-analytic diffeomorphisms of the circle, Ergodic Theory Dynam. Systems, 22 (2002), 835-844. doi: 10.1017/S014338570200041X.  Google Scholar

[7]

J. Franks and M. Handel, Entropy zero area preserving diffeomorphisms of $S^2$, Geometry & Topology, 16 (2012), 2187-2284. doi: 10.2140/gt.2012.16.2187.  Google Scholar

[8]

J. Franks, Generalizations of the Poincaré-Birkhoff Theorem, Ann. of Math. (2), 128 (1988), 139-151. doi: 10.2307/1971464.  Google Scholar

[9]

J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99-107. doi: 10.1017/S0143385700009366.  Google Scholar

[10]

N. V. Ivanov, Subgroups of Teichmüller Modular Groups, Translated from the Russian by E. J. F. Primrose and revised by the author, Translations of Mathematical Monographs, 115, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[11]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[12]

A. Katok, Hyperbolic measures and commuting maps in low dimension, Discrete Contin. Dynam. Systems, 2 (1996), 397-411. doi: 10.3934/dcds.1996.2.397.  Google Scholar

[13]

J. McCarthy, A "Tits-alternative'' for subgroups of surface mapping class groups, Trans. Amer. Math. Soc., 291 (1985), 583-612. doi: 10.2307/2000100.  Google Scholar

[14]

C. P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math., 26 (1974), 187-200. doi: 10.1007/BF01418948.  Google Scholar

[15]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[16]

S. Sternberg, Local $C^n$ transformations of the real line, Duke Math. J., 24 (1957), 97-102. doi: 10.1215/S0012-7094-57-02415-8.  Google Scholar

[17]

J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. doi: 10.1016/0021-8693(72)90058-0.  Google Scholar

show all references

References:
[1]

M. Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991.  Google Scholar

[2]

M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out($F^n$). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2), 151 (2000), 517-623. doi: 10.2307/121043.  Google Scholar

[3]

M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out($F^n$). II. A Kolchin type theorem, Ann. of Math. (2), 161 (2005), 1-59. doi: 10.4007/annals.2005.161.1.  Google Scholar

[4]

E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42.  Google Scholar

[5]

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc., 91 (1984), 503-504. doi: 10.2307/2045329.  Google Scholar

[6]

B. Farb and P. Shalen, Groups of real-analytic diffeomorphisms of the circle, Ergodic Theory Dynam. Systems, 22 (2002), 835-844. doi: 10.1017/S014338570200041X.  Google Scholar

[7]

J. Franks and M. Handel, Entropy zero area preserving diffeomorphisms of $S^2$, Geometry & Topology, 16 (2012), 2187-2284. doi: 10.2140/gt.2012.16.2187.  Google Scholar

[8]

J. Franks, Generalizations of the Poincaré-Birkhoff Theorem, Ann. of Math. (2), 128 (1988), 139-151. doi: 10.2307/1971464.  Google Scholar

[9]

J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99-107. doi: 10.1017/S0143385700009366.  Google Scholar

[10]

N. V. Ivanov, Subgroups of Teichmüller Modular Groups, Translated from the Russian by E. J. F. Primrose and revised by the author, Translations of Mathematical Monographs, 115, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[11]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[12]

A. Katok, Hyperbolic measures and commuting maps in low dimension, Discrete Contin. Dynam. Systems, 2 (1996), 397-411. doi: 10.3934/dcds.1996.2.397.  Google Scholar

[13]

J. McCarthy, A "Tits-alternative'' for subgroups of surface mapping class groups, Trans. Amer. Math. Soc., 291 (1985), 583-612. doi: 10.2307/2000100.  Google Scholar

[14]

C. P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math., 26 (1974), 187-200. doi: 10.1007/BF01418948.  Google Scholar

[15]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[16]

S. Sternberg, Local $C^n$ transformations of the real line, Duke Math. J., 24 (1957), 97-102. doi: 10.1215/S0012-7094-57-02415-8.  Google Scholar

[17]

J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. doi: 10.1016/0021-8693(72)90058-0.  Google Scholar

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