On some planar Baumslag-Solitar actions.

A form of geometric rigidity for actions of the groups BS(1, n) = ⟨a, b | aba−1 = b n⟩ is discussed, with applications to its actions by homeomorphisms of the torus. The setting considered here is representations by orientation-preserving homeomorphisms of the plane under the assumption that a is linear and b has bounded displacement. The main results show a range of rigidity behaviours of the representation if the action of a is diagonalizable over the reals (depending on the eigenvalues and excluding eigenvalues of modulus one) and different rigidities in the parabolic and elliptic cases. The applications to representations by homeomorphisms of the torus (obtained from the rigidity results on the plane by lifting) include a new proof that there are no faithful actions of BS(1, n) for which the action of a is an Anosov map with stretch factor greater than n, shown earlier by J. Alonso et al. [Discrete Contin. Dyn. Syst. 35, No. 5, 1817–1827 (2015; Zbl 1328.37025)] and a proof that there is no such faithful action for which the action of a is a Dehn twist map.