保持粗中位数的自同构
Coarse-median preserving automorphisms
AI总结：
本文引入保持粗中位数的自同构，刻画 RAAG/RACG 不动子群的几何性质，并将无限阶外自同构实现为有限秩中位数空间上的同位似。
摘要：
本文有三个主要目标。首先，我们研究直角 Artin 群和直角 Coxeter 群的自同构的不动子群。如果 φ 是 RAAG 的非扭曲自同构，或 RACG 的任意自同构，我们证明 Fix φ 是有限生成且无畸变的。在将 φ 替换为其某个幂之后，我们证明 Fix φ 关于标准词度量是拟凸的。这意味着 Fix φ 是可分的，并且是 Haglund–Wise 意义下的特殊群。相比之下，RAAG 存在一些“扭曲的”自同构，其 Fix φ 无畸变但不是 F 型（因而不是特殊的），或是 F 型但有畸变，甚至是无限生成的。其次，我们引入粗中位数群的“保持粗中位数”自同构概念，它在上述结果中起关键作用。我们证明，RAAG 的自同构保持粗中位数，当且仅当它是非扭曲的。另一方面，Gromov 双曲群和直角 Coxeter 群的所有自同构都保持粗中位数。这些事实还给出了 RAAG 和 RACG 的 Nielsen 实现的新的或更初等的证明。最后，我们证明，对于每个特殊群 G（Haglund–Wise 意义下），G 的每个无限阶、保持粗中位数的外自同构都可实现为有限秩中位数空间 X 的同位似，而 X 配备了“适度的”等距 G-作用。这推广了 Paulin 的经典结果：双曲群 H 的每个无限阶外自同构都会射影地稳定一棵小 H-树。
英文摘要：
This paper has three main goals. First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If $φ$ is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that ${\rm Fix}~φ$ is finitely generated and undistorted. Up to replacing $φ$ with a power, we show that ${\rm Fix}~φ$ is quasi-convex with respect to the standard word metric. This implies that ${\rm Fix}~φ$ is separable and a special group in the sense of Haglund-Wise. By contrast, there exist "twisted" automorphisms of RAAGs for which ${\rm Fix}~φ$ is undistorted but not of type $F$ (hence not special), of type $F$ but distorted, or even infinitely generated. Secondly, we introduce the notion of "coarse-median preserving" automorphism of a coarse median group, which plays a key role in the above results. We show that automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving. These facts also yield new or more elementary proofs of Nielsen realisation for RAAGs and RACGs. Finally, we show that, for every special group $G$ (in the sense of Haglund-Wise), every infinite-order, coarse-median preserving outer automorphism of $G$ can be realised as a homothety of a finite-rank median space $X$ equipped with a "moderate" isometric $G$-action. This generalises the classical result, due to Paulin, that every infinite-order outer automorphism of a hyperbolic group $H$ projectively stabilises a small $H$-tree.
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