Burnside problem for groups of homeomorphisms of compact surfaces

A group $Gamma$ is said to be periodic if for any $g$ in $Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $mathrm{O}(3,R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose of a previous paper of the authors) is finite.

Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms

A group $G$ is said to be periodic if for any $gin G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $mu$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.

$C^1$-actions of Baumslag-Solitar groups on $S^1$

Let $BS(1, n)=< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ ngeq 2$. It is known that B(1, n) is isomorphic to the group generated by the two affine maps of the line : $f_0(x) = x + 1$ and $h_0(x) = nx $. The action on $S^1 = RR cup {infty}$ generated by these two affine maps $f_0$ and $h_0 $ is called the standard affine one. We prove that any representation of BS(1,n) into $Diff^1(S^1)$ is (up to a finite index subgroup) semiconjugated to the standard affine action.

Actions of Baumslag-Solitar groups on surfaces

Let $BS(1,n) =< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ ngeq 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.