Stallings' Group is Simply Connected at Infinity

Abstract

Let $F_2$ be the free group on two generators and let $B_n$ ($n\ge 2$) denote the kernel of the homomorphism $$F_2\times \cdots (n)\cdots \times F_2\to \mathbb{Z}$$ sending all generators to the generator $1$ of $\mathbb{Z}$. The groups $B_k$ are called the {\it Bieri-Stallings} groups and $B_k$ is type ${\mathcal{F}}_{k-1}$ but not ${\mathcal{F}}_k$. For $n\ge 3$ there are short exact sequences of the form $$1\to B_{n-1}\to B_n\to F_2\to 1.$$ This exact sequence can be used to show that $B_n$ is $(n-3)$-connected at infinity for $n\ge 3$. Stallings' proved that $B_2$ is finitely generated but not finitely presented. We conjecture that for $n\ge 2$, $B_n$ is $(n-2)$-connected at infinity. For $n=2$, this means that $B_2$ is 1-ended and for $n=3$ that $B_3$ (typically called Stallings' group) is simply connected at infinity. We verify the conjecture for $n=2$ and $n=3$. Our main result is the case $n=3$: Stalling's group is simply connected at $\infty$.

18 pages 10 figures