Actions of Baumslag-Solitar groups on surfaces

Abstract

Let $BS(1,n)=<a,b|ab{a}^{-1}=b^n>$ be the solvable Baumslag-Solitar group, where $n\ge 2$. 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 $\TT ^2$, 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 $<f,h|h\circ f\circ h^{-1}=f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= \TT^2$, 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 $\TT^2$. 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 $C^1$ then $fix(f)$ contains any minimal set.

We clarified and improved some results and we added examples