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May  2013, 33(5): 1945-1964. doi: 10.3934/dcds.2013.33.1945

Actions of Baumslag-Solitar groups on surfaces

Nancy Guelman 1, and  Isabelle Liousse 2,

1. 

IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30,Montevideo
2. 

Laboratoire Paul PAINLEVÉ, Université de Lille1, 59655 Villeneuve d'Ascq Cédex, France

Received  November 2011 Revised  July 2012 Published  December 2012

Let $BS(1, n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that $BS(1, n)$ is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $.
    This paper deals with the dynamics of actions of $BS(1, n)$ on closed orientable surfaces. We exhibit a smooth $BS(1,n)$-action without finite orbits on $\mathbb{T} ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid.
    We develop a general dynamical study for faithful topological $BS(1,n)$-actions on closed surfaces $S$. We prove that such actions $ < f, h \ | \ h o f o h^{-1} = f^n >$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty.
    When $S= \mathbb{T}^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of $BS(1,n)$ on $\mathbb{T}^2$.
    When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ and isotopic to identity then $fix(f)$ contains any minimal set.
Keywords: Baumslag Solitar group, Actions on surfaces, minimal sets..
Mathematics Subject Classification: Primary: 37C85, 37E30; Secondary: 37B0.
Citation: Nancy Guelman, Isabelle Liousse. Actions of Baumslag-Solitar groups on surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945
References:
[1]

M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1.  doi: 10.1007/PL00004317.  Google Scholar

[2]

C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61.  doi: 10.2307/1971485.  Google Scholar

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L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877.  doi: 10.2140/gt.2004.8.877.  Google Scholar

[4]

S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075.   Google Scholar

[5]

B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001).   Google Scholar

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B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.  doi: 10.1215/S0012-7094-01-10636-4.  Google Scholar

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J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107.  doi: 10.2307/2001018.  Google Scholar

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J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441.  doi: 10.1215/S0012-7094-06-13132-0.  Google Scholar

[9]

J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557.  doi: 10.1017/S0143385706001088.  Google Scholar

[10]

J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443.  doi: 10.3934/jmd.2007.1.443.  Google Scholar

[11]

E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.   Google Scholar

[12]

N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701.  doi: 10.2140/agt.2011.11.1701.  Google Scholar

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M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993).   Google Scholar

[14]

M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975).   Google Scholar

[15]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977).   Google Scholar

[16]

E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97.   Google Scholar

[17]

A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395.  doi: 10.1090/S0002-9939-09-10173-9.  Google Scholar

[18]

M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490.  doi: 10.1112/jlms/s2-40.3.490.  Google Scholar

[19]

Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399.   Google Scholar

[20]

A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13.  doi: 10.1007/s00574-004-0002-2.  Google Scholar

[21]

J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149.  doi: 10.1017/S0143385700003345.  Google Scholar

[22]

J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567.   Google Scholar

[23]

J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123.   Google Scholar

[24]

M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273.   Google Scholar

[25]

M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218.  doi: 10.1007/BFb0076436.  Google Scholar

show all references

References:
[1]

M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1.  doi: 10.1007/PL00004317.  Google Scholar

[2]

C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61.  doi: 10.2307/1971485.  Google Scholar

[3]

L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877.  doi: 10.2140/gt.2004.8.877.  Google Scholar

[4]

S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075.   Google Scholar

[5]

B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001).   Google Scholar

[6]

B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.  doi: 10.1215/S0012-7094-01-10636-4.  Google Scholar

[7]

J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107.  doi: 10.2307/2001018.  Google Scholar

[8]

J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441.  doi: 10.1215/S0012-7094-06-13132-0.  Google Scholar

[9]

J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557.  doi: 10.1017/S0143385706001088.  Google Scholar

[10]

J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443.  doi: 10.3934/jmd.2007.1.443.  Google Scholar

[11]

E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.   Google Scholar

[12]

N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701.  doi: 10.2140/agt.2011.11.1701.  Google Scholar

[13]

M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993).   Google Scholar

[14]

M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975).   Google Scholar

[15]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977).   Google Scholar

[16]

E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97.   Google Scholar

[17]

A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395.  doi: 10.1090/S0002-9939-09-10173-9.  Google Scholar

[18]

M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490.  doi: 10.1112/jlms/s2-40.3.490.  Google Scholar

[19]

Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399.   Google Scholar

[20]

A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13.  doi: 10.1007/s00574-004-0002-2.  Google Scholar

[21]

J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149.  doi: 10.1017/S0143385700003345.  Google Scholar

[22]

J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567.   Google Scholar

[23]

J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123.   Google Scholar

[24]

M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273.   Google Scholar

[25]

M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218.  doi: 10.1007/BFb0076436.  Google Scholar

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