ACTIONS OF BAUMSLAG-SOLITAR GROUPS ON SURFACES

Let BS(1,n) =< a, b | aba(-1) = b(n) > be the solvable Baumslag-Solitar group, where n >= 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 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 circle f circle 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 = 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 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 then fix(f) contains any minimal set.