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 [J. Topol. Anal. 12 (2020), pp. 713–734], whose original proof contains a serious gap that currently lacks a resolution.
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Sergio Zamora (2025) studied this question.
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