  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.
