Profiniteness of higher rank volume

TL;DR

This paper demonstrates that the covolume of certain lattices in higher rank Lie groups and specific hyperbolic manifolds can be uniquely determined by their profinite completions, revealing new invariance properties.

Contribution

It establishes that covolume is a profinite invariant for irreducible lattices with CSP in higher rank groups and for octonionic hyperbolic manifolds without CSP.

Findings

Covolume is determined by the profinite completion for lattices with CSP.

Volume is a profinite invariant for octonionic hyperbolic manifolds.

Results extend understanding of invariants in geometric group theory.

Abstract

We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds.