Linear progress in fibres (Q6594558)

Let \(G\) a finitely generated group and \(H \leq G\) a finitely generated subgroup of \(G\). For any choice of proper word metrics on \(H\) and \(G\), the inclusion of \(H\) into \(G\) is a Lipschitz map. However, distances are contracted by arbitrary amounts by the inclusion, this can be quantified by the distortion function, the smallest function bounding the word norm of \(H\) in terms of the word norm of \(G\). Many examples from low-dimensional topology exhibit exponential distortion a well-known example is the fundamental groups of fibres in fibred hyperbolic 3-manifolds (see [\textit{B. Deroin} et al., Mosc. Math. J. 9, No. 2, 263--303 (2009; Zbl 1193.37034)]).\N\NLet \(\Sigma\) denote a closed orientable surface of genus \(g > 2\). The first result in the paper under review is Theorem 1.1: Let \(1 \rightarrow \pi_{1}(\Sigma) \rightarrow \Gamma \rightarrow Q \rightarrow 1\) be a hyperbolic group extension of a closed surface group. Then, there is a constant \(c>0\) that depends only on the word metrics so that almost every geodesic ray in \(\pi_{1}(\Sigma)\) (sampled by the Lebesgue measure under an identification of \(\pi_{1}(\Sigma)\) with \(\mathbb{H}^{2}\)) makes linear progress with speed at least \(c\) in the word metric on \(\Gamma\).\N\NTheorem 1.2 and Theorem 1.3 follow the same vein but their statements are too technical to be reported here. They extend Theorem 1.1 respectively to the universal curve over a Teichmüller disc and to the extension induced by the Birman exact sequence.