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 $MCG(S)$.

中文翻译：

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图上的紧束树和表面束的模型几何

我们将复杂测地线的概念推广到复杂树木的曲线中。然后，我们使用紧密树来为图形上的某些表面束构造模型几何。这扩展了由Brock，Canary和Minsky提出的双简并双曲3形流形在终结层​​定理的证明过程中的组合模型的某些方面。因此，我们获得配备了几何的一致Gromov-双曲几何模型空间$G$动作，在哪里 $G$ 接受表格的确切顺序

$$\mathrm{1个}\to {\pi }_{\mathrm{1个}}（\mathrm{小号}）\to G\to 问\to \mathrm{1个}。$$

这里 $\mathrm{小号}$ 是属的封闭表面 $G>\mathrm{1个}$ 和 $问$ 属于映射类组的一类特殊的自由凸协紧子组 $\mathrm{中号}CG（\mathrm{小号}）$。