# American Institute of Mathematical Sciences

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. [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. [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. [4] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. [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. [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. [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. [8] J. Franks, Generalizations of the Poincaré-Birkhoff Theorem, Ann. of Math. (2), 128 (1988), 139-151. doi: 10.2307/1971464. [9] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99-107. doi: 10.1017/S0143385700009366. [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. [11] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [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. [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. [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. [15] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. [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. [17] J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. doi: 10.1016/0021-8693(72)90058-0.

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##### References:
 [1] M. Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. [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. [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. [4] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. [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. [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. [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. [8] J. Franks, Generalizations of the Poincaré-Birkhoff Theorem, Ann. of Math. (2), 128 (1988), 139-151. doi: 10.2307/1971464. [9] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99-107. doi: 10.1017/S0143385700009366. [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. [11] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [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. [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. [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. [15] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. [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. [17] J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. doi: 10.1016/0021-8693(72)90058-0.
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