Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields

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.

💡 Research Summary

The paper establishes a linear upper bound on the total Betti numbers of finite‑volume negatively curved orbifolds with coefficients in an arbitrary field. Extending Gromov’s 1985 theorem, which was limited to torsion‑free lattices, and Samet’s 2013 result, which required characteristic 0, the authors prove that for any Hadamard manifold (X) with sectional curvature (-1\le K<0) and any lattice (\Gamma\subset\operatorname{Isom}(X)), there exists a constant (C_n) depending only on the dimension such that

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