Tight trees and model geometries of surface bundles over graphs

Abstract

We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly degenerate hyperbolic 3-manifolds developed by Brock, Canary, and Minsky during the course of their proof of the Ending Lamination Theorem. Thus we obtain uniformly Gromov-hyperbolic geometric model spaces equipped with geometric $G-$actions, where $G$ admits an exact sequence of the form $$1 \to π_1(S) \to G \to Q \to 1.$$ Here $S$ is a closed surface of genus $g > 1$ and $Q$ belongs to a special class of free convex cocompact subgroups of the mapping class group $MCG(S)$.