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Register a new user$C^1$-actions of Baumslag-Solitar groups on $S^1$
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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.
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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.
We give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation. The set of equivalence classes carries a natural partial order first introduced by Abbott-Balasubramanya-Osin, and we describe the resulting poset completely. There are finitely many equivalence classes of actions, and each equivalence class contains the action on a point, a tree, or the hyperbolic plane.
Denote by $DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^infty$ manifold $M$. We show that if $dim(M)geq2$ and $r eq dim(M)+1$, then any homomorphism from $DC(M)_0$ to ${Diff}^1(R)$ or ${Diff}^1(S^1)$ is trivial.
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.
In this paper we classify Baumslag-Solitar groups up to commensurability. In order to prove our main result we give a solution to the isomorphism problem for a subclass of Generalised Baumslag-Solitar groups.
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