The conformal boundary and the boundary of the convex core (Q1847794)

Let a hyperbolic 3-manifold \(N\) be the quotient of \(\mathbb{H}^3\) by a group of isometries \(\Gamma\) and \(\Omega(\Gamma)\) be the domain of discontinuity of \(\Gamma\). Consider \(\partial_C N = \Omega(\Gamma) / \Gamma\) to be the conformal boundary at infinity of \(N\). The author investigates the relationship between the conformal boundary at infinity of a hyperbolic 3-manifold and the boundary of its convex core. The main theorem of the paper gives the following nice estimate. Suppose that \(N\) is a hyperbolic 3-manifold and \(r : \partial_C N \to \partial C(N)\) is the nearest point retraction from its conformal boundary to the boundary of its convex core. If \(\alpha\) is a closed curve in the conformal boundary of length \(L\), then \(l_{\partial C(N)} (r(\alpha)^*)< 45 L e^{L/2}\), where \(l_{\partial C(N)} (r(\alpha)^*)\) denotes the length of the closed geodesic in the intrinsic metric on \(\partial C(N)\) in the homotopy class of \(r(\alpha)\).