On the strong Liouville property of covering spaces

Polymerakis, Panagiotis

doi:10.1007/s11118-022-10019-8

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On the strong Liouville property of covering spaces

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Polymerakis, Panagiotis
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s11118-022-10019-8
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Polymerakis, P. (in press). On the strong Liouville property of covering spaces. Potential Analysis, Published Online - Print pending. doi:10.1007/s11118-022-10019-8.

Cite as: https://hdl.handle.net/21.11116/0000-000A-AC9D-4

Abstract

Lyons and Sullivan conjectured in Lyons and Sullivan (J. Differential Geom. 19(2), 299–323, 1984) that if p : M → N is a normal Riemannian covering, with N closed, and M has exponential volume growth, then there are non-constant, positive harmonic functions on M. This was proved recently in Polymerakis (Adv. Math. 379, 107552–107558, 2021) exploiting the Lyons-Sullivan discretization and some sophisticated estimates on the green metric on groups. In this note, we provide a self-contained proof relying only on elementary properties of the Brownian motion.