ACTIONS OF SOLVABLE BAUMSLAG-SOLITAR GROUPS ON SURFACES WITH (PSEUDO)-ANOSOV ELEMENTS

Let BS(1, n) = [a,b : aba(-1) = b(n)] be the solvable Baumslag-Solitar group, where n >= 2. We study representations of BS(1, n) by homeomorphisms of closed surfaces of genus g >= 1 with (pseudo)-Anosov elements. That is, we consider a closed surface S of genus g >= 1, and homeomorphisms f, h : S --> S such that hfh(-1) = f(n), for some n >= 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 [f, h] 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).