Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

DOI10.1215/00127094-2021-0051arXiv2001.03377WikidataQ115517625 ScholiaQ115517625MaRDI QIDQ6332623

Hee Oh, Sam Edwards

Publication date: 10 January 2020

Abstract: Let $$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(mathrm{T}^1(mathcalM))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(mathcalM)$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $mathcalM$ when $Gamma$ is a Zariski dense subgroup contained in an arithmetic subgroup of $mathrm{SO}^{circ}(d+1,1)$.

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Discrete subgroups of Lie groups (22E40) Spectral theory; trace formulas (e.g., that of Selberg) (11F72) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40)

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