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