From the boundary of the convex core to the conformal boundary (Q1868440)

Let \(N\) be an orientable hyperbolic 3-manifold with finitely generated fundamental group. The boundary \(\partial C(N)\) of the convex core \(C(N)\) of \(N\) and its conformal boundary \(\partial_cN\) are homeomorphic finite area hyperbolic surfaces. A beautiful result of Sullivan states that there exists a uniform constant \(K\) such that if \(\partial C(N)\) is incompressible in the convex core \(C(N)\), then there is a \(K\)-bi-Lipschitz homeomorphism between \(\partial_cN\) and \(\partial C(N)\). This homeomorphism is defined by using a ``nearest point retraction map''. In the paper under review, the authors give a partial generalization of this result of Sullivan, by assuming only that \(N\) has finitely generated fundamental group. They show that in this more general setting, the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse, and they give bounds of the Lipschitz constants of these maps in terms of the length of the shortest compressible curve on the conformal boundary.