Semistability at $\infty$, $\infty$-ended groups and group cohomology

Abstract:

A finitely presented group $G$, is semistable at $\infty$ if for some (equivalently any) finite complex $X$, with ${\pi _1}(X) = G$, any two proper maps $r, s:[0, \infty ) \to \tilde X$ ($\equiv$ the universal cover of $X$) that determine the same end of $\tilde X$ are properly homotopic in $\tilde X$. If $G$ is semistable at $\infty$, then ${H^2}(G; ZG)$ is free abelian. $0$- and $2$-ended groups are all semistable at $\infty$. Theorem. If $G = A{{\ast }_C}B$ where $C$ is finite and $A$ and $B$ are finitely presented, semistable at $\infty$ groups, then $G$ is semistable at $\infty$. Theorem. If $\alpha :C \to D$ is an isomorphism between finite subgroups of the finitely presented semistable at $\infty$ group $H$, then the resulting $HNN$ extension is semistable at $\infty$. Combining these results with the accessibility theorem of M. Dunwoody gives Theorem. If all finitely presented $1$-ended groups are semistable at $\infty$, then all finitely presented groups are semistable at $\infty$.