  A 1-ended finitely presented group has semistable fundamental group at
$\infty$ if it acts geometrically on some (equivalently any) simply connected
and locally finite complex $X$ with the property that any two proper rays in
$X$ are properly homotopic. If $G$ has semistable fundamental group at $\infty$
then one can unambiguously define the fundamental group at $\infty$ for $G$.
The problem, asking if all finitely presented groups have semistable
fundamental group at $\infty$ has been studied for over 40 years. If $G$ is an
ascending HNN extension of a finitely presented group then indeed, $G$ has
semistable fundamental group at $\infty$, but since the early 1980's it has
been suggested that the finitely presented groups that are ascending HNN
extensions of {\it finitely generated} groups may include a group with
non-semistable fundamental group at $\infty$. Ascending HNN extensions
naturally break into two classes, those with bounded depth and those with
unbounded depth. Our main theorem shows that bounded depth finitely presented
ascending HNN extensions of finitely generated groups have semistable
fundamental group at $\infty$. Semistability is equivalent to two weaker
asymptotic conditions on the group holding simultaneously. We show one of these
conditions holds for all ascending HNN extensions, regardless of depth. We give
a technique for constructing ascending HNN extensions with unbounded depth.
This work focuses attention on a class of groups that may contain a group with
non-semistable fundamental group at $\infty$.
