Out（$\mathbb{F}$）中的相对不可约自由子群
Relatively irreducible free subroups in Out($\mathbb{F}$)
摘要：
本文证明了：给定秩为$\geq 3$的有限秩自由群$\mathbb{F}$和两个指数增长的外自同构$ψ$和$ϕ$，以及与它们相关联的对偶叠层对$Λ^\pm_ψ$和$Λ^\pm_ϕ$，并给定一个自由因子系统$\mathcal{F}$，其余边数$\geq 2$，$ϕ, ψ$均保持$\mathcal{F}$，使得对$(ϕ, Λ^\pm_ϕ), (ψ, Λ^\pm_ψ)$相对于$\mathcal{F}$独立，则存在$\exists$ $M\geq 1$，使得对于任何整数$m,n \geq M$，群$\langle ϕ^m, ψ^n \rangle$是秩为2的自由群，其所有非平凡元素除了可能的$ϕ, ψ$的幂及其共轭之外，相对于$\mathcal{F}$是完全不可约的，其中层压对相对于$\mathcal{F}$填充。此外，如果两个$Λ^\pm_ϕ, Λ^\pm_ψ$都是非几何的，则该叠片对也是非几何的。我们还证明了在某些自然条件下，由这类子群诱导的扩张群是相对双曲的。
英文摘要：
We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $ψ$ and $ϕ$ with dual lamination pairs $Λ^\pm_ψ$ and $Λ^\pm_ϕ$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\geq 2$, $ϕ, ψ$ each preserving $\mathcal{F}$, so that the pair $(ϕ, Λ^\pm_ϕ), (ψ, Λ^\pm_ψ)$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\geq 1$, such that for any integer $m,n \geq M$, the group $\langle ϕ^m, ψ^n \rangle$ is a free group of rank 2, all of whose non-trivial elements except perhaps the powers of $ϕ, ψ$ and their conjugates, are fully irreducible relative to $\mathcal{F}$ with a lamination pair which fills relative to $\mathcal{F}$. In addition if both $Λ^\pm_ϕ, Λ^\pm_ψ$ are non-geometric then this lamination pair is also non-geometric. We also prove that the extension groups induced by such subgroups will be relatively hyperbolic under some natural conditions.
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