Incoherence of free-by-free and surface-by-free groups

Incoherence and fibering of many free-by-free groups

We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $F_2\rtimes F_n$ to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including free-by-free, surface-by-surface, and finitely generated-by-RAAG, algebraically fiber if they have excessive homology.

自由群自同构群的高阶非凝聚性

Higher incoherence of the automorphism groups of a free group

本研究证明自由群自同构群$\operatorname{Aut}(F_n)$和外自同构群$\operatorname{Out}(F_n)$存在满足特定有限型条件的非$m$-凝聚子群，得到紧的上界，并借助$\operatorname{Aut}(F_2)$子群的不等价扩张完成证明。

凝聚RFRS群

Coherent RFRS groups

本文研究上同调维数至多2的有限生成虚拟RFRS群的凝聚性等价条件，据此刻画二维Coxeter群等类群的凝聚性并证实相关猜想，附录中解决了立方化情形下的相关公开问题。

Higher dimensional algebraic fiberings for pro-$p$ groups

We prove some conditions for higher dimensional algebraic fibering of pro-$p$ group extensions and we establish corollaries about incoherence of pro-$p$ groups. In particular, if $G = K \rtimes Γ$ is a pro-$p$ group, $Γ$ a finitely generated free pro-$p$ group with $d(Γ) \geq 2$, $K$ a finitely presented pro-$p$ group with $N$ a normal pro-$p$ subgroup of $K$ such that $K/ N \simeq \mathbb{Z}_p$ and $N$ not finitely generated as a pro-$p$ group, then $G$ is incoherent (in the category of pro-$p$ groups). Furthermore we show that if $K$ is a free pro-$p$ group with $d(K) = 2$ then either $Aut_0…

Random graphs of free groups contain surface subgroups

A random graph of free groups contains a surface subgroup

On surface subgroups of doubles of free groups

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.