Using subnormality to show the simple connectivity at infinity of a finitely presented group

Abstract:

A CW-complex $X$ is simply connected at infinity if for each compact $C$ in $X$ there exists a compact $D$ in $X$ such that loops in $X - D$ are homotopically trivial in $X - C$. Let $G$ be a finitely presented group and $X$ a finite CW-complex with fundamental group $G$. $G$ is said to be simply connected at infinity if the universal cover of $X$ is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if $G$ and a normal subgroup $H$ are infinite finitely presented groups with $G/H$ infinite and either $H$ or $G/H$ $1$-ended, then $G$ is simply connected at infinity. In the case where $H$ is $1$-ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on $H$ cannot be reduced to "finitely generated." We address the question: if $N$ is normal in $H$ and $H$ is normal in $G$ and these are infinite groups with $N$ and $G$ finitely presented and either $N$ or $G/H$ is $1$-ended, is $G$ simply connected at infinity? In the case that $N$ is $1$-ended, the answer is shown to be yes. In the case that $G/H$ is $1$-ended, we exhibit a class of such groups that are not simply connected at infinity.