# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945
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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.
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