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, (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. 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. 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.

[5]

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set,, Proc. Amer. Math. Soc., 91 (1984), 503. 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. doi: 10.1017/S014338570200041X.

[7]

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

[8]

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

[9]

J. Franks, Recurrence and fixed points of surface homeomorphisms,, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99. 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, (1992).

[11]

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

[12]

A. Katok, Hyperbolic measures and commuting maps in low dimension,, Discrete Contin. Dynam. Systems, 2 (1996), 397. 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. 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. doi: 10.1007/BF01418948.

[15]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. 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. doi: 10.1215/S0012-7094-57-02415-8.

[17]

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

show all references

References:
[1]

M. Artin, Algebra,, Prentice Hall, (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. 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. 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.

[5]

M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set,, Proc. Amer. Math. Soc., 91 (1984), 503. 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. doi: 10.1017/S014338570200041X.

[7]

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

[8]

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

[9]

J. Franks, Recurrence and fixed points of surface homeomorphisms,, Ergodic Theory Dynam. Systems, 8$^*$ (1988), 99. 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, (1992).

[11]

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

[12]

A. Katok, Hyperbolic measures and commuting maps in low dimension,, Discrete Contin. Dynam. Systems, 2 (1996), 397. 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. 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. doi: 10.1007/BF01418948.

[15]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. 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. doi: 10.1215/S0012-7094-57-02415-8.

[17]

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

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