Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo-)Anosov elements

Abstract

Let $BS(1,n)=<a,b:ab{a}^{-1}=b^n>$ be the solvable Baumslag-Solitar group, where $n\ge 2$. We study representations of $BS(1,n)$ by homeomorphisms of closed surfaces with (pseudo-)Anosov elements. That is, we consider a closed surface $S$, and homeomorphisms $f,h:S\to S$ such that $hf{h}^{-1}=f^n$, for some $n\ge 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is pseudo-Anosov with stretch factor $\lambda >1$. We show that $<f,h>$ is not a faithful representation of $BS(1,n)$ if $\lambda >n$. Moreover, we show that there are no faithful representations of $BS(1,n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$).

new version with improvements

References in corpus (2)

• Groups acting on manifolds: around the Zimmer program
• $C^1$-actions of Baumslag-Solitar groups on $S^1$