Surface group actions and length bounds

Let Σ be compact orientable surface, and write Γ = π1(Σ). Let H be a hyperbolic space (in the sense of Gromov [Gr]) admitting a discrete action by Γ. and write M = H/Γ. Let X(Σ) be the set of homotopy classes of essential non-peripheral closed curves in Σ. Let G(Σ) be the curve graph with vertex set V (G) = X(Σ) (i.e. the 1-skeleton of the curve complex [H]). To each γ ∈ X(Σ) we can associate a “stable length” l M (γ). In [MaM2], the authors define the notion of a “tight geodesic” in G(Σ). The aim of this paper is to describe certain bounds on the stable lengths of curves arising in a tight geodesic. These are a direct generalisation of the “a-priori bounds” described by Minsky in [Mi], where H is hyperbolic 3-space. Here we generalise the approach in [Bo2]. One of the main motivations of this work is its use in describing “coarse models” of hyperbolic 3-manifolds, see [Bo4]. If H is simply connected, we can think of this in terms of the quotient M = H/Γ. Then l M (γ) is, up to additive constant, equal to the length of a shortest (or nearly shortest) representative of the free homotopy class of γ in M . In fact, there is no essential loss of generality in assuming H to be simply connected, as we will explain later. To be more precise, we make an assumption on the space H, namely (H1) which corresponds to coarse bounded geometry, and a hypothesis on the action, namely (H2) which substitutes for the Margulis lemma. (It is effectively the conclusion of the Margulis lemma if H happens to be a simply connected manifold of pinched negative curvature.) Recall that a subset, Q ⊆ H is r-separated if d(x, y) ≥ r for all distinct x, y ∈ Q. We assume: