Actions of Baumslag-Solitar groups on surfaces

Details Actions of Baumslag-Solitar groups on surfaces Actions of Baumslag-Solitar groups on surfaces

2010-04-13

arxiv.org/abs/1004.2126v2

Mathematics

Dynamical Systems

We clarified and improved some results and we added examples

Scientific paper

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 $\TT ^2$, 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 $$ 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.

Affiliated with

Also associated with

No associations

Rating

Actions of Baumslag-Solitar groups on surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Actions of Baumslag-Solitar groups on surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Actions of Baumslag-Solitar groups on surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-281877