AI 中文总结
本文研究了相对自由分裂与自由因子复形的大尺度几何及其作用动力学，证明了相关复形的超几何性并探讨了外自同构作用的动力学特性。
AI 中文摘要
对于任何群Γ和任何自由因子系统A of Γ，相对外自同构群Out(Γ;A)自然地作用于相对自由分裂复形FS(Γ;A)和相对自由因子系统复形FF(Γ;A)，推广了已知的Out(F_n)对绝对自由分裂复形FS(F_n)和绝对自由因子复形F(F_n)的作用。本文概述了三部分工作，涉及FS(Γ;A)和FF(Γ;A)的大尺度几何以及Out(Γ;A)元素对这些复形的作用的几何动力学。在第一部分arXiv:1407.3508中，证明了FS(Γ;A)和FF(Γ;A)的超几何性。在第二和第三部分arXiv:2212.09907、arXiv:2503.07532中，研究了Out(Γ;A)元素的几何动力学与其相对列车轨道代表的动态之间的关系。第二部分的主要工具是“Two Over All Theorem”，用于表达Stallings折叠路径在FS(Γ;A)中的指数膨胀性质。第三部分的主要工具是“填充路径”，用于阐述和证明“Two Over All Theorem”的强版本。
英文摘要
For any group $Γ$ and any free factor system $\mathscr A$ of $Γ$, the relative outer automorphism group $\text{Out}(Γ;\mathscr A)$ acts naturally on the relative free splitting complex $\mathcal{F\!S}(Γ;\mathscr A)$ and on the complex of relative free factor systems $\mathcal{F\!F}(Γ;\mathscr A)$, generalizing the well known actions of $\text{Out}(F_n)$ on the absolute free splitting complex $\mathcal{F\!S}(F_n)$ and the absolute free factor complex ${\mathcal{F}}(F_n)$ of the rank $n$ free group $F_n$. This overview summarizes a three part work regarding the large scale geometry of $\mathcal{F\!S}(Γ;\mathscr A)$ and $\mathcal{F\!F}(Γ;\mathscr A)$ and the geometric dynamics of the actions on these complexes by elements of $\text{Out}(Γ;\mathscr A)$. In Part I arXiv:1407.3508 we prove hyperbolicity of $\mathcal{F\!S}(Γ;\mathscr A)$ and of $\mathcal{F\!F}(Γ;\mathscr A)$. In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of $\text{Out}(Γ;\mathscr A)$ and the dynamics of its relative train track representatives. The main tool in Part II is the Two Over All Theorem, expressing an exponential flaring property of Stallings fold paths in $\mathcal{F\!S}(Γ;\mathscr A)$. The main tools in Part III are "filling paths", used to formulate and prove a strong version of the Two Over All Theorem.
Comments21 pages (+ 4 pages of references). In Version 2, fixed Question 7 regarding a conjectural hyperbolic $\text{Aut}(Γ;\mathscr A)$ complex
