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

Actions of Baumslag-Solitar groups on surfaces

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.
Citation: Nancy Guelman, Isabelle Liousse. Actions of Baumslag-Solitar groups on surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945
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show all references

References:
[1]

Preprint, Prepub. IRMA Lille, 34 (1994), 1-16. doi: 10.1007/PL00004317.  Google Scholar

[2]

Ann. of Math., 129 (1989), 61-69. doi: 10.2307/1971485.  Google Scholar

[3]

Geom. Topol., 8 (2004), 877-924. doi: 10.2140/gt.2004.8.877.  Google Scholar

[4]

Ann. Inst. Fourier (Grenoble), 52 (2002), 1075-1091.  Google Scholar

[5]

Preprint (2001). Google Scholar

[6]

Duke Math. J., 106 (2001), 581-597. doi: 10.1215/S0012-7094-01-10636-4.  Google Scholar

[7]

Trans. Amer. Math. Soc., 311 (1989), 107-116. doi: 10.2307/2001018.  Google Scholar

[8]

Duke Math. J., 131 (2006), 441-468. doi: 10.1215/S0012-7094-06-13132-0.  Google Scholar

[9]

Erg. Th. and Dyn. Sys., 27 (2007), 1557-1581. doi: 10.1017/S0143385706001088.  Google Scholar

[10]

Jour. of Modern Dynamics, 1 (2007), 443-464. doi: 10.3934/jmd.2007.1.443.  Google Scholar

[11]

Enseign. Math. (2), 47 (2001), 329-407.  Google Scholar

[12]

Algebraic & Geometric Topology, 11 (2011), 1701-1707. doi: 10.2140/agt.2011.11.1701.  Google Scholar

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in "Geometric Group Theory, London Math. Soc. LNS 182 Cambridge University Press, Cambridge" 2 (1993).  Google Scholar

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Lecture Notes in Math. Springer-Verlag, 468 (1975). Google Scholar

[15]

Lecture Notes in Math. Springer-Verlag, Berlin-New York, 583 (1977).  Google Scholar

[16]

Comment. Math. Helv., 39 (1964), 97-110.  Google Scholar

[17]

Proc. Amer. Math. Soc., 138 (2010), 1395-1403. doi: 10.1090/S0002-9939-09-10173-9.  Google Scholar

[18]

J. London Math. Soc., 40 (1989), 490-506. doi: 10.1112/jlms/s2-40.3.490.  Google Scholar

[19]

Hokkaido Math. Jour., 23 (1994), 399-422.  Google Scholar

[20]

Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13-50. doi: 10.1007/s00574-004-0002-2.  Google Scholar

[21]

Ergod. Th. and Dynam. Sys., 6 (1986), 149-161. doi: 10.1017/S0143385700003345.  Google Scholar

[22]

Comment. Math. Helv., 51 (1976), 567-584.  Google Scholar

[23]

Mosc. Math. J., 3 (2003), 123-171.  Google Scholar

[24]

Global Analysis Proceedings of the Symposium on Pure Mathematics, Amer. Math. Soc., Providence, XIV (1970), 273-276.  Google Scholar

[25]

Iteration Theory and Its Functional Equations, Lecture Notes, 1163 (1985), 218-231. doi: 10.1007/BFb0076436.  Google Scholar

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