Volume is a profinite invariant of higher rank locally symmetric spaces

Is the volume of a locally symmetric space determined by the profinite completion of its fundamental group?

We show that this is the case in higher rank: the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. It is an open question whether a similar result holds for hyperbolic 3-manifolds. We explain how our methods generalize to another rank-one situation: octonionic hyperbolic congruence manifolds.

(This is based on joint work with Holger Kammeyer and Ralf Köhl)