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Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence

Luca F. Di Cerbo, Mark Stern

Abstract

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.

Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence

Abstract

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.

Paper Structure

This paper contains 8 sections, 23 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Dimension Estimates Revisited
  3. Negative Ricci Curvature
  4. Pinched Negative Sectional Curvature
  5. Uniformly Discrete Sequences
  6. Thick-Thin Decomposition Revisited
  7. Pointwise Bounds for Harmonic Forms on Manifolds with Boundary
  8. Non Uniformly Discrete Sequences

Key Result

Theorem 5

Let $\{\Gamma_l\}_l$ be a sequence of uniformly discrete, torsion free lattices acting co-compactly on a symmetric space $G/K$ of non-compact type. Let $\{\Gamma_l\backslash G/K\}_l$ be the associated sequence of compact locally symmetric spaces. For any $k\leq\dim(G/K)$, if $\{\Gamma_l\backslash G/ where the $k$-th $L^2$-Betti number of the symmetric space, $\beta^{(2)}_{k}(G/K)$, is defined in B

Theorems & Definitions (44)

  • Definition 3
  • Theorem 5: Corollary 1.4 in Bergeron
  • Theorem 6: Theorem 1.8. in Bergeron
  • Definition 7
  • Remark 8
  • Theorem 10
  • Theorem 11
  • Lemma 15
  • proof
  • Lemma 16
  • ...and 34 more