Title: Non-cocompact Group Actions and $π_1$-Semistability at Infinity,
Abstract: A finitely presented 1-ended group $G$ has {\it semistable fundamental group
at infinity} if $G$ acts geometrically on a simply connected and locally
compact ANR $Y$ having the property that any two proper rays in $Y$ are
properly homotopic. This property of $Y$ captures a notion of connectivity at
infinity stronger than "1-ended", and is in fact a feature of $G$, being
independent of choices. It is a fundamental property in the homotopical study
of finitely presented groups. While many important classes of groups have been
shown to have semistable fundamental group at infinity, the question of whether
every $G$ has this property has been a recognized open question for nearly
forty years. In this paper we attack the problem by considering a proper {\it
but non-cocompact} action of a group $J$ on such an $Y$. This $J$ would
typically be a subgroup of infinite index in the geometrically acting
over-group $G$; for example $J$ might be infinite cyclic or some other subgroup
whose semistability properties are known. We divide the semistability property
of $G$ into a $J$-part and a "perpendicular to $J$" part, and we analyze how
these two parts fit together. Among other things, this analysis leads to a
proof (in a companion paper) that a class of groups previously considered to be
likely counter examples do in fact have the semistability property.