Pro-p groups and towers of rational homology spheres

TL;DR

This paper proves that certain hyperbolic 3-manifolds constructed via pro-p groups are rational homology spheres unconditionally, and provides criteria for towers of such manifolds to have zero first Betti number at each level.

Contribution

It establishes unconditionally that specific hyperbolic 3-manifolds are rational homology spheres and offers a pro-p group theoretical criterion for towers to have zero first Betti number.

Findings

Manifolds are rational homology spheres unconditionally.

Provides a criterion for towers to have zero first Betti number.

Uses purely pro-p group theoretical methods.

Abstract

In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3-manifolds to have first Betti number 0 at each level. The methods involved are purely pro-p group theoretical.