Abstract
We show that asymptotically the first Betti number of a Shimura curve satisfies the Gauss–Bonnet equality where is hyperbolic volume; equivalently where is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is . This generalizes previous results obtained by Frączyk, on which we rely, and uses the same main tool, namely Benjamini–Schramm convergence.
Citation
Mikołaj Frączyk. Jean Raimbault. "Betti numbers of Shimura curves and arithmetic three-orbifolds." Algebra Number Theory 13 (10) 2359 - 2382, 2019. https://doi.org/10.2140/ant.2019.13.2359
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