Geometrization of Piecewise homeomorphisms of the circle

TL;DR

This paper investigates the conditions under which certain piecewise circle homeomorphisms can be realized as Bowen-Series maps linked to surface groups, connecting geometric and dynamical properties.

Contribution

It characterizes the geometric and dynamical criteria needed for realizing piecewise homeomorphisms as Bowen-Series-like maps for surface groups.

Findings

Identifies sufficient conditions for realization as Bowen-Series maps

Links geometric properties to dynamical realizability

Provides a framework for understanding circle homeomorphisms in geometric group theory

Abstract

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 ${\rm Homeo}^+ (S^1)$?