Semistability of amalgamated products, HNN-extensions, and all   one-relator groups
  The authors announce the following theorem.   Theorem 1. If $G=A*_H B$ is an amalgamated product where $A$ and $B$ are finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity. If $G=A*_H$ is an HNN-extension where $A$ is finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity. 
Michael L. Mihalik, Steven T. Tschantz