For any torus with Ricci curvature at least -(n-1) and diameter at most D, some finite cover has volume and diameter bounded by constants depending only on n and D.
Stability of tori under lower sectional curvature
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Torus covers with controlled volume and diameter
For any torus with Ricci curvature at least -(n-1) and diameter at most D, some finite cover has volume and diameter bounded by constants depending only on n and D.