3-Manifold Seminar: RFRS for congruence covers

Seminar | April 23 | 3:40-5 p.m. | 736 Evans Hall

 Ian Agol, UC Berkeley

 Department of Mathematics

This is an update on a similar talk given a couple of years ago. Baker and Reid asked a question about whether a tower of principal congruence covers of an arithmetic hyperbolic 3-manifold associated to powers of a prime ideal might satisfy the RFRS condition (Residually Finite Rational Solvable). A positive answer to this question implies the existence of a congruence cover that fibers over the circle. We prove that this (essentially) follows if the first principal congruence subgroup is p-torsion free, where the prime ideal divides p. Examples are known for some Bianchi groups, and now for O(4,1;Z) and for a certain complex hyperbolic surface. The O(4,1;Z) example gives infinitely many Bianchi groups with discriminant a sum of two squares having a fibered congruence cover. Joint with Matthew Stover.