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

Juan Alonso; Nancy Guelman; Juliana Xavier

doi:10.3934/dcds.2015.35.1817

Article Contents

2015, Volume 35, Issue 5: 1817-1827. Doi: 10.3934/dcds.2015.35.1817

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

1.
Centro de Matemática, Facultad de Ciencias, Iguá 4225, Montevideo, CP 11400
2.
IMERL, Facultad de Ingeniería, Julio Herrera y Reissig 565, Montevideo, CP 11300

Received: February 28, 2014

Revised: August 31, 2014

Published: May 2017

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Abstract

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces of genus $g\geq 1$ with (pseudo)-Anosov elements. That is, we consider a closed surface $S$ of genus $g\geq 1$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 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 $\langle f,h \rangle$ is not a faithful representation of $BS(1, n)$ if $\lambda > n$. We also 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 $\lambda$).

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Mathematics Subject Classification: Primary: 37C85, 37E30, 37B05, 37E45.

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