TY - JOUR

T1 - A random cover of a compact hyperbolic surface has relative spectral gap 316-ε

AU - Magee, Michael

AU - Naud, Frédéric

AU - Puder, Doron

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2022/6

Y1 - 2022/6

N2 - Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature - 1. For each n∈ N, let Xn be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or Xn is an eigenvalue of the associated Laplacian operator Δ X or ΔXn. We say that an eigenvalue of Xn is new if it occurs with greater multiplicity than in X. We prove that for any ε> 0 , with probability tending to 1 as n→ ∞, there are no new eigenvalues of Xn below 316-ε. We conjecture that the same result holds with 316 replaced by 14.

AB - Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature - 1. For each n∈ N, let Xn be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or Xn is an eigenvalue of the associated Laplacian operator Δ X or ΔXn. We say that an eigenvalue of Xn is new if it occurs with greater multiplicity than in X. We prove that for any ε> 0 , with probability tending to 1 as n→ ∞, there are no new eigenvalues of Xn below 316-ε. We conjecture that the same result holds with 316 replaced by 14.

UR - http://www.scopus.com/inward/record.url?scp=85130482794&partnerID=8YFLogxK

U2 - 10.1007/s00039-022-00602-x

DO - 10.1007/s00039-022-00602-x

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85130482794

SN - 1016-443X

VL - 32

SP - 595

EP - 661

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -