Semistability at the end of a group extension

A $1$-ended ${\text{CW}}$-complex, $Q$, is semistable at $\infty$ if all proper maps $r:\ [0,\infty) \to Q$ are properly homotopic. If ${X_1}$ and ${X_2}$ are finite ${\text{CW}}$-complexes with isomorphic fundamental groups, then the universal cover of ${X_1}$ is semistable at $\infty$ if and only if the universal cover of ${X_2}$ is semistable at $\infty$. Hence, the notion of a finitely presented group being semistable at $\infty$ is well defined. We prove

Main Theorem. Let $1 \to H \to G \to K \to 1$ be a short exact sequence of finitely generated infinite groups. If $G$ is finitely presented, then $G$ is semistable at $\infty$.

Theorem. If $A$ and $B$ are locally compact, connected noncompact $CW$-complexes, then $A \times B$ is semistable at $\infty$.

Theorem. $\langle\;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle$ is semistable at $\infty$.

The proofs are geometrical in nature and the main tool is covering space theory.