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Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields

Guy Kapon, Raz Slutsky

TL;DR

We extend Gromov's linear Betti-number bound from manifolds to finite-volume negatively curved orbifolds with coefficients in arbitrary fields, proving $\sum_{i=1}^n b_i(X/\Gamma;\mathbb{F}) \le C_n\,\mathrm{Vol}(X/\Gamma)$. A key ingredient is a uniform bound on the homology of spherical quotients: for any finite group $G$ acting linearly on $S^k$, $b_i(S^k/G;\mathbb{F})$ is bounded by a constant depending only on $k$ (e.g., $C_k = 3^k(k+1)!^{\log(3k)}$). The argument combines a thick-thin decomposition in the orbifold setting, a Morse-theoretic analysis of the displacement function, and the uniform spherical-quotient bound to control singular contributions in arbitrary characteristic. This yields linear growth in volume for mod-$p$ and other coefficients, with potential extensions to broader non-positively curved orbifolds and implications for homology growth problems in arithmetic and geometric group theory.

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.

Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields

TL;DR

We extend Gromov's linear Betti-number bound from manifolds to finite-volume negatively curved orbifolds with coefficients in arbitrary fields, proving . A key ingredient is a uniform bound on the homology of spherical quotients: for any finite group acting linearly on , is bounded by a constant depending only on (e.g., ). The argument combines a thick-thin decomposition in the orbifold setting, a Morse-theoretic analysis of the displacement function, and the uniform spherical-quotient bound to control singular contributions in arbitrary characteristic. This yields linear growth in volume for mod- and other coefficients, with potential extensions to broader non-positively curved orbifolds and implications for homology growth problems in arithmetic and geometric group theory.

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 . 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 to arbitrary characteristic. The key new input is a quantitative bound on the homology of spherical quotients.
Paper Structure (17 sections, 16 theorems, 66 equations)

This paper contains 17 sections, 16 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Finite Group Actions on Spheres
  3. Linear actions of finite groups on spheres
  4. Jordan reduction to a bounded extension of an abelian group
  5. A uniform bound in the abelian case
  6. A characteristic-$p$ bound for finite CW complexes
  7. The cyclic case
  8. From $C_p$ to $p$--groups
  9. General finite groups via transfer
  10. Combining with the Jordan reduction
  11. The quasi-thick part of non-positively curved orbifolds
  12. Setting, displacement, and quasi-thick parts
  13. A good cover of the quasi-thick part
  14. Bundles over a cover
  15. The Morse function and critical pairs
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a constant $C_n$, depending only on the dimension, such that for every Hadamard manifold $X$ and torsion-free lattice $\Gamma$ where $b_i(X / \Gamma)$ are the betti numbers with respect to coefficients in any field.

Theorems & Definitions (32)

  • Theorem 1.1: Gromov, 1985
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more