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 $\mathrm{PSL}(2,\mathbb{Q}^{\mathtt{ac}})$--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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2021
- DOI:
- 10.48550/arXiv.2105.01022
- arXiv:
- arXiv:2105.01022
- Bibcode:
- 2021arXiv210501022L
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Group Theory;
- Mathematics - Number Theory;
- Primary 57M50;
- Secondary 57M10;
- 30F40;
- 20E18
- E-Print:
- 39 pages