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