Geometrization of Piecewise homeomorphisms of the circle

In this work we study the following realization problem: given a piecewise homeomorphism $\mathrm{\Phi }:S^1\to S^1$, which geometrical and dynamical conditions on $\mathrm{\Phi }$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}^{+}(S^1)$?