Sur le spectre et la topologie des vari\'et\'es hyperboliques de congruence : les cas complexe et quaternionien

TL;DR

This paper extends spectral and topological results for complex and quaternionic hyperbolic manifolds, providing optimal spectral bounds and a Lefschetz property for arithmetic quotients, building on prior foundational work.

Contribution

It generalizes spectral and topological results to complex and quaternionic hyperbolic manifolds, achieving optimal bounds and extending Lefschetz properties.

Findings

Optimal spherical spectrum results for complex and quaternionic hyperbolic manifolds

Extension of Lefschetz property to arithmetic quotients of complex balls

Generalization of Nair's theorem with an optimal version

Abstract

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex balls. This generalizes a recent theorem of Arvind Nair and gives an optimal version of it.