Manifold classification from the descriptive viewpoint

We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set-theoretic point of view. This work is largely foundational in conception and character, recording both a framework for general study and Borel complexity computations for some of the most fundamental classes of manifolds. We show, for example, that for all $n\ge 0$, the homeomorphism problem for compact topological $n$-manifolds is Borel equivalent to the relation ${=}_{\mathbb{N}}$ of equality on the natural numbers, while the homeomorphism problem for noncompact topological $2$-manifolds is of maximal complexity among equivalence relations classifiable by countable structures. A nontrivial step in the latter consists of proving Borel measurable formulations of the Jordan–Schoenflies and surface triangulation theorems. Turning our attention to groups and geometric structures, we show, strengthening results of Stuck–Zimmer and Andretta–Camerlo–Hjorth, that the conjugacy relation on discrete subgroups of any noncompact semisimple Lie group is essentially countable universal. So too, as a corollary, is the isometry relation for complete hyperbolic $n$-manifolds for any $n\ge 2$, generalizing a result of Hjorth–Kechris. We then show that the isometry relation for complete hyperbolic $n$-manifolds with finitely generated fundamental group is, in contrast, Borel equivalent to the equality relation ${=}_{\mathbb{R}}$ on the real numbers when $n=2$, but that it is not concretely classifiable when $n=3$; thus there exists no Borel assignment of numerical complete invariants to finitely generated Kleinian groups up to conjugacy. We close with a survey of the most immediate open questions.