Quasi-convexity and shrinkwrapping (Q1048472)

Let \(S\) be a closed surface of genus \(>1\), \(N\) be an oriented hyperbolic 3-manifold and \(j :S\rightarrow N\) be a \(\pi_1\)-injective map. Let us denote by \(C(B,j)\) the set of homotopy classes of noncontractible simple loops \(\alpha\) where \(j(\alpha)\) has a representative of length \(\leq B\) in \(N\). This is a subspace of the complex of curves \(C(S)\) of \(S\). The following theorem was proved in [\textit{Y. N. Minsky}, Invent. Math. 146, No.~1, 143--192 (2001; Zbl 1061.37026)]. Theorem. For \(B\) large enough depending on \(\chi(S)\), the set \(C(B,j)\) is \(K\)-quasi-convex in \(C(S)\), with \(K\) depending only on \(\chi(S)\) and \(B\). (this means that for \(p,q\in C(B,j)\) every geodesic connecting them is in the \(K\)-neighborhood of \(C(B,j)\)). The aim of the paper under review is to extend the preceding theorem. In particular the condition about \(\pi_1\)-injectivity of \(j\) is replaced by a weaker condition. Such a result is seen as a step to understand the following question, addressed in the paper under review: \textbf{Question} Do there exist constants \(B,K\) such that, for every irreducible representation \(\rho : \pi_1(S)\rightarrow \mathrm{PSL}_2(\mathbb{C})\), \(C(B,\rho)\) is \(K\)-quasi-convex in \(C(S)\)?