# Tight trees and model geometries of surface bundles over graphs

## TL;DR

This paper introduces tight trees to develop new geometric models for surface bundles over graphs, extending prior work on hyperbolic 3-manifolds and providing hyperbolic spaces with group actions related to surface groups and convex cocompact subgroups.

## Contribution

It generalizes tight geodesics to tight trees and constructs model geometries for surface bundles over graphs, extending the combinatorial models of hyperbolic 3-manifolds.

## Key findings

- Constructed Gromov-hyperbolic model spaces with geometric group actions.
- Extended combinatorial models to include surface bundles over graphs.
- Connected the models to convex cocompact subgroups of the mapping class group.

## 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 $$1 \to \pi_1(S) \to G \to Q \to 1.$$ 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 $MCG(S)$.