Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields

TL;DR

This paper establishes a linear bound on the Betti numbers of finite-volume negatively curved orbifolds over any field, extending Gromov's theorem from manifolds to orbifolds and answering a question by Samet.

Contribution

It generalizes Gromov's linear Betti number bound from manifolds to orbifolds in negative curvature for arbitrary coefficient fields.

Findings

Betti numbers grow at most linearly with volume

Extension of Gromov's theorem to orbifolds

Answer to Samet's question in arbitrary characteristic

Abstract

We show that the Betti numbers of finite-volume negatively curved orbifolds grow at most linearly with the volume, with coefficients in an arbitrary field. In particular, this gives a linear bound for the Betti numbers of finite-volume hyperbolic orbifolds over $\mathbb{F}_p$. This extends a theorem of Gromov from manifolds to orbifolds in negative curvature, and answers a question of Samet, by strengthening his theorem from characteristic $0$ to arbitrary characteristic. The key new input is a quantitative bound on the homology of spherical quotients.