Rigidity of Almost-Isometric Universal Covers

TL;DR

This paper investigates the rigidity of universal covers under almost-isometries, showing that certain geometric properties like volume growth entropy are preserved, and establishing conditions where no almost-isometry exists.

Contribution

It introduces new rigidity results for universal covers under almost-isometries, extending understanding of geometric invariance in Riemannian geometry.

Findings

Almost-isometries preserve volume growth entropy.

Under specific conditions, no almost-isometry exists between certain universal covers.

Rigidity results apply to various geometric settings.

Abstract

Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.