Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements.

Let BS(1,n)=⟨a,b:aba−1=bn⟩ be the solvable Baumslag-Solitar group, where n≥2 whose origin goes back to [G. Baumslag and D. Solitar, Bull. Am. Math. Soc. 68, 199–201 (1962; Zbl 0108.02702)].
The authors study representations of BS(1,n) by homeomorphisms of closed surfaces of genus g≥1 with (pseudo)-Anosov elements. Thus, they consider a closed surface S of genus g≥1, and homeomorphisms f,h:S→S such that hfh−1=fn, 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 λ>1. The paper shows that ⟨f,h⟩ is not a faithful representation of BS(1,n) if λ>n. As a consequence, 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 λ).
The present results are related to those in [N. Guelman and I. Liousse, Discrete Contin. Dyn. Syst. 33, No. 5, 1945–1964 (2013; Zbl 1311.37014)], working in the C1 category.