图的可分离同调与可分离复形
Separable homology of graphs and the separability complex
AI总结：
本文研究了图的可分离同调性质，通过引入可分离复形，证明其连通性与fundamental群生成条件的关系，并展示了该复形的非超几何特性。
摘要：
我们引入了可分离复形，这是一个与有限正则覆盖的玫瑰相关的一复形，并证明它连通当且仅当关联覆盖的fundamental群由其与proper free factors of F_n的交集生成。如果关联的覆盖对应于F_n的一个特征子群，则可分离复形允许由Out(F_n)通过等距作用。我们证明玫瑰的可分离复形有无限直径且非超几何，从而表明它不与自由分裂复形或自由因子复形准等距。作为结果，我们得到F_n的Cayley图，其生成集由所有primitive elements of F_n组成，是非超几何的。
英文摘要：
We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The separability complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of $\mathbf{F}_n$ with generating set consisting of all primitive elements of $\mathbf{F}_n$ is nonhyperbolic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
