曲面乘自由群的立方化
Cubulating Surface-by-free Groups
AI总结：
本文研究了闭曲面基本群与自由群构成的正合序列，提供了双曲群可立方化的充分条件并构造实例，主要结果为非拟凸合并子群下虚拟特殊双曲群的组合定理。
摘要：
设 $$1 \ o H \ o G \ o Q \ o 1$$ 为正合序列，其中 $H=π_1(S)$ 是亏格大于一的闭曲面 $S$ 的基本群，$G$ 是双曲群，$Q$ 是有限生成自由群。本文旨在提供充分条件以证明 $G$ 是可立方化的，并构造满足这些条件的例子。主要结果可视为当合并子群不是拟凸时的虚拟特殊双曲群的组合定理。其要素包括轨道理论、Wise的拟凸层级定理、Masur-Minsky基于子曲面投影的映射类群距离估计，以及用于证明结尾叶理定理的双退化Kleinian曲面群模型几何。本文附录由Manning、Mj和Sageev撰写，通过证明 $G$ 的可立方化可由具有基本群 $G$ 的图上曲面丛中本质不可压缩拟凸轨道的存在性推导得出，从而证明了一个归约定理。
英文摘要：
Let $$1 \to H \to G \to Q \to 1$$ be an exact sequence where $H= π_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide sufficient conditions to prove that $G$ is cubulable and construct examples satisfying these conditions. The main result may be thought of as a combination theorem for virtually special hyperbolic groups when the amalgamating subgroup is not quasiconvex. Ingredients include the theory of tracks, the quasiconvex hierarchy theorem of Wise, the distance estimates in the mapping class group from subsurface projections due to Masur-Minsky and the model geometry for doubly degenerate Kleinian surface groups used in the proof of the ending lamination theorem. An appendix to this paper by Manning, Mj, and Sageev proves a reduction theorem by showing that cubulability of $G$ follows from the existence of an essential incompressible quasiconvex track in a surface bundle over a graph with fundamental group $G$.
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