Torus covers with controlled volume and diameter

TL;DR

This paper proves that under certain curvature and diameter constraints, a torus has a finite covering with controlled volume and diameter, partially addressing a previous result with an unresolved gap.

Contribution

It provides a new proof of a covering space property for tori under curvature and diameter bounds, addressing gaps in prior work.

Findings

Existence of finite coverings with bounded volume and diameter

Partial recovery of a previous theorem by Kloeckner and Sabourau

Addresses gaps in earlier proofs regarding torus coverings

Abstract

We show that under a lower Ricci curvature bound and an upper diameter bound, a torus admits a finite-sheeted covering space with volume bounded from below and diameter bounded from above. This partially recovers a result of Kloeckner and Sabourau, whose original proof contains a serious gap that currently lacks a resolution.