  Stackability for finitely presented groups consists of a dynamical system
that iteratively moves paths into a maximal tree in the Cayley graph. Combining
with formal language theoretic restrictions yields auto- or algorithmic
stackability, which implies solvability of the word problem. In this paper we
give two new characterizations of the stackable property for groups, and use
these to show that every HNN extension of a stackable group is stackable. We
apply this to exhibit a wide range of Dehn functions that are admitted by
stackable and autostackable groups, as well as an example of a stackable group
with unsolvable word problem. We use similar methods to show that there exist
finitely presented metabelian groups that are non-constructible but admit an
autostackable structure.
