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Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David Borthwick, Colin Guillarmou (2016)

Journal of the European Mathematical Society

On geometrically finite hyperbolic manifolds Γ d , including those with non-maximal rank cusps, we give upper bounds on the number N ( R ) of resonances of the Laplacian in disks of size R as R . In particular, if the parabolic subgroups of Γ satisfy a certain Diophantine condition, the bound is N ( R ) = 𝒪 ( R d ( log R ) d + 1 ) .

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