Actions of   Baumslag-Solitar groups   on  surfaces

Nancy Guelman; Isabelle Liousse

doi:10.3934/dcds.2013.33.1945

Article Contents

2013, Volume 33, Issue 5: 1945-1964. Doi: 10.3934/dcds.2013.33.1945

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Actions of Baumslag-Solitar groups on surfaces

Nancy Guelman^1, and
Isabelle Liousse^2,

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

Received: November 2011

Revised: July 2012

Published: May 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C85, 37E30; Secondary: 37B05.

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