Actions of   Baumslag-Solitar groups   on  surfaces

Nancy Guelman; Isabelle Liousse

doi:10.3934/dcds.2013.33.1945

Article Contents

2013, Volume 33, Issue 5: 1945-1964. Doi: 10.3934/dcds.2013.33.1945

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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

Received: October 31, 2011

Revised: June 30, 2012

Published: April 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C85, 37E30; Secondary: 37B05.

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References

[1]	M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes, Preprint, Prepub. IRMA Lille, 34 (1994), 1-16.doi: 10.1007/PL00004317.
[2]	C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$, Ann. of Math., 129 (1989), 61-69.doi: 10.2307/1971485.
[3]	L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$, Geom. Topol., 8 (2004), 877-924.doi: 10.2140/gt.2004.8.877.
[4]	S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075-1091.
[5]	B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups, Preprint (2001).
[6]	B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke Math. J., 106 (2001), 581-597.doi: 10.1215/S0012-7094-01-10636-4.
[7]	J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-116.doi: 10.2307/2001018.
[8]	J. Franks and M. Handel, Distortion elements in group actions on surfaces, Duke Math. J., 131 (2006), 441-468.doi: 10.1215/S0012-7094-06-13132-0.
[9]	J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$, Erg. Th. and Dyn. Sys., 27 (2007), 1557-1581.doi: 10.1017/S0143385706001088.
[10]	J. Franks, Handel and K. Parwani, Fixed points of abelian actions, Jour. of Modern Dynamics, 1 (2007), 443-464.doi: 10.3934/jmd.2007.1.443.
[11]	E. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.
[12]	N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$, Algebraic & Geometric Topology, 11 (2011), 1701-1707.doi: 10.2140/agt.2011.11.1701.
[13]	M. Gromov, Asymptotic invariants of infinite groups, in "Geometric Group Theory, London Math. Soc. LNS 182 Cambridge University Press, Cambridge" 2 (1993).
[14]	M. Hirsch, A stable analytic foliation with only exceptional minimal set, Lecture Notes in Math. Springer-Verlag, 468 (1975).
[15]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math. Springer-Verlag, Berlin-New York, 583 (1977).
[16]	E. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv., 39 (1964), 97-110.
[17]	A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups, Proc. Amer. Math. Soc., 138 (2010), 1395-1403.doi: 10.1090/S0002-9939-09-10173-9.
[18]	M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc., 40 (1989), 490-506.doi: 10.1112/jlms/s2-40.3.490.
[19]	Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line, Hokkaido Math. Jour., 23 (1994), 399-422.
[20]	A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13-50.doi: 10.1007/s00574-004-0002-2.
[21]	J. F. Plante, Fixed points of Lie group actions on surfaces, Ergod. Th. and Dynam. Sys., 6 (1986), 149-161.doi: 10.1017/S0143385700003345.
[22]	J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv., 51 (1976), 567-584.
[23]	J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$, Mosc. Math. J., 3 (2003), 123-171.
[24]	M. Shub, Expanding maps, Global Analysis Proceedings of the Symposium on Pure Mathematics, Amer. Math. Soc., Providence, XIV (1970), 273-276.
[25]	M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow, Iteration Theory and Its Functional Equations, Lecture Notes, 1163 (1985), 218-231.doi: 10.1007/BFb0076436.