New Piecewise Linear representations of surface groups in Homeo(S¹)

Resumen: Piecewise Linear representations of surface groups, i.e. in PL_λ(S¹) in Homeo(S¹), appeared in many works starting with results by Ghys, Matsumoto, and more recently in Minakawa. These constructions are based on some dynamical properties of the geodesic flow.

Here, the idea is to consider the surface group as an abstract group given by some presentations and use some other dynamical properties, the one of some Bowen-Series-like maps.

The factors λ that appear in these constructions are algebraic integers, as in the previous constructions, but are related to the growth rate of the group presentations rather than to some growth of a pseudo-Anosov cross section to a flow. This raises some funny questions