  Autostackability for finitely presented groups is a topological property of
the Cayley graph combined with formal language theoretic restrictions, that
implies solvability of the word problem. The class of autostackable groups is
known to include all asynchronously automatic groups with respect to a
prefix-closed normal form set, and all groups admitting finite complete
rewriting systems. Although groups in the latter two classes all satisfy the
homological finiteness condition $FP_\infty$, we show that the class of
autostackable groups includes a group that is not of type $FP_3$. We also show
that the class of autostackable groups is closed under graph products and
extensions.
