短层级双曲群 I：不可数多个粗中值结构
Short hierarchically hyperbolic groups I: uncountably many coarse median structures
AI总结：
本文证明五穿孔球面映射类群及某些右角Artin群具有不可数多个粗中值结构，通过发展短层级双曲群理论并利用拟态射构造拟直线实现。
摘要：
我们证明，具有五个穿孔的球面的映射类群允许不可数多个粗等变粗中值结构。对于其定义图是连通的、无三角形且无正方形、且至少具有三个顶点的右角Artin群，我们也证明了同样的结论。值得注意的是，在后一种情况下，我们产生的粗中值结构并非由余紧立方化诱导。为了获得上述结果，我们发展了短层级双曲群（HHG）的理论，该理论还包括大型和双曲型Artin群、图流形群以及Veech群的扩张。我们开发了修改其层级结构的工具，包括使用拟态射来构造作为坐标空间的拟直线，而这正是粗中值结构丰富性的来源。这些技术具有独立的意义，并在与Alessandro Sisto合作的后续论文中被用于研究短HHG的商。在此过程中，我们还澄清了Hagen、Martin和Sisto关于大型和双曲型Artin群层级双曲性的一个证明。
英文摘要：
We prove that the mapping class group of a sphere with five punctures admits uncountably many coarsely equivariant coarse median structures. The same is shown for right-angled Artin groups whose defining graphs are connected, triangle- and square-free, and have at least three vertices. Remarkably, in the latter case, the coarse median structures we produce are not induced by cocompact cubulations. To obtain the above results, we develop the theory of short hierarchically hyperbolic groups (HHG), which also include Artin groups of large and hyperbolic type, graph manifold groups, and extensions of Veech groups. We develop tools to modify their hierarchical structure, including using quasimorphisms to construct quasilines that serve as coordinate spaces, and this is where the abundance of coarse median structures comes from. These techniques are of independent interest, and are used in a follow-up paper with Alessandro Sisto to study quotients of short HHG. In the process, we also clarify a proof of Hagen, Martin, and Sisto on hierarchical hyperbolicity of Artin groups of large and hyperbolic type.
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