Finite Quotients, Arithmetic Invariants, and Hyperbolic Volume

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 $\text {PSL}(2,{\mathbb {Q}}^{{\textrm{ac}}})$-representations of their fundamental groups, up to conjugacy; moreover, corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras. (Here, ${\mathbb {Q}}^{{\textrm{ac}}}$ 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 $\text {PSL}(2,{\mathbb {C}})$ with isomorphic profinite completions have identical invariant trace fields, isomorphic invariant quaternion algebras, identical covolume, and identical arithmeticity.

Funding

National Outstanding Youth Science Fund Project of National Natural Science Foundation of China(11925101)

Key Technologies Research and Development Program(2020YFA0712800)

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