A remark on torsion growth in homology and volume of 3-manifolds

TL;DR

This paper discusses the relationship between torsion growth in homology, the volume of 3-manifolds, and the implications of Lück's conjecture, suggesting that identical finite quotients imply equal volumes.

Contribution

It establishes a connection between Lück's conjecture and the volume equality of 3-manifolds based on their fundamental groups' finite quotients.

Findings

Lück's conjecture implies volume equality for 3-manifolds with identical finite quotients.

The paper links torsion growth in homology to geometric properties of 3-manifolds.

It provides a new perspective on the relationship between algebraic and geometric invariants.

Abstract

We show that L\"uck's conjecture on torsion growth in homology implies that two 3-manifolds have equal volume if the fundamental groups have the same set of finite quotients.