标题： 图中自由群的循环边群的虚拟同调挠 显示英文标题

标题： Virtual homological torsion in graphs of free groups with cyclic edge groups

摘要： 设$G$是一个在自由群与循环边群的图上分裂的双曲群。我们证明，除非$G$同构于自由群和曲面群的自由积，否则每个有限交换群$M$都会作为某个有限指数子群$G'\le G$的交换化中的直和因子出现。作为应用，我们推断出自由群和曲面群的自由积在具有循环边群的双曲群的图上是拟同构刚性的。我们还得出结论，在自由群中的部分曲面词由它们在有限群上诱导的字测度确定。 显示英文摘要

摘要： Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in the abelianization of some finite-index subgroup $G'\le G$. As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.