  We first consider three well-known chain conditions in the space of marked
groups: the minimal condition on centralizers, the maximal condition on
subgroups, and the maximal condition on normal subgroups. For each condition,
we produce a characterization in terms of well-founded
descriptive-set-theoretic trees. Using these characterizations, we demonstrate
that the sets given by these conditions are co-analytic and not Borel in the
space of marked groups. We then adapt our techniques to show elementary
amenable marked groups may be characterized by well-founded
descriptive-set-theoretic trees, and therefore, elementary amenability is
equivalent to a chain condition. Our characterization again implies the set of
elementary amenable groups is co-analytic and non-Borel. As corollary, we
obtain a new, non-constructive, proof of the existence of finitely generated
amenable groups that are not elementary amenable.
