Finite quotients, arithmetic invariants, and hyperbolic volume

TL;DR

This paper establishes a deep connection between the profinite completions of hyperbolic 3-manifold groups and their arithmetic invariants, linking algebraic representations to geometric properties under certain conjectural assumptions.

Contribution

It demonstrates that isomorphisms of profinite completions induce correspondences between representations and invariants of hyperbolic 3-manifolds, advancing understanding of their arithmetic and geometric structures.

Findings

Profinite isomorphisms correspond to bijections between Zariski dense representations.

Corresponding representations share invariant trace fields and quaternion algebras.

Under conjectural assumptions, profinite isomorphisms imply identical volume and arithmeticity.

Abstract

For any pair of orientable closed hyperbolic $3$--manifolds, this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense $\mathrm{PSL}(2,\mathbb{Q}^{\mathtt{ac}})$--representations of their fundamental groups, up to conjugacy; moreover, corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras. (Here, $\mathbb{Q}^{\mathtt{ac}}$ denotes an algebraic closure of $\mathbb{Q}$.) Next, assuming the $p$--adic Borel regulator injectivity conjecture for number fields, this paper shows that uniform lattices in $\mathrm{PSL}(2,\mathbb{C})$ with isomorphic profinite completions have identical invariant trace fields, isomorphic invariant quaternion algebras, identical covolume, and identical arithmeticity.