Transverse shear, rotation sets, and loxodromic torus homeomorphisms

We prove that in many cases, the composition of two torus homeomorphisms with thin rotation sets produces a rotation set with non-empty interior. This has interesting consequences on the geometry of subgroups of Homeo_0(T^2) and their action on the fine curve graph: we get new "Tits alternatives", that is, rigidity results for subgroups not containing a Schottky group, and results on the structure for purely elliptic groups. All the results are based on a single quantitative transverse stretching theorem: if f "stretches" much more than g in one direction, and g stretches much more than f in the orthogonal direction, then the rotation set of their composition contains an explicit rectangle.

Joint work with C. Freijo, S. Hansel and F. Le Roux.