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Generalized Graph Manifolds, Residual Finiteness, and the Singer Conjecture

Luca F. Di Cerbo, Michael Hull

Abstract

We prove the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups. In real dimension three, where a result of Hempel ensures that the fundamental group is always residually finite, we then provide a Price type inequality proof of a well-known result of Lott and Lueck. Finally, we give several classes of higher graph manifolds which do indeed have residually finite fundamental groups.

Generalized Graph Manifolds, Residual Finiteness, and the Singer Conjecture

Abstract

We prove the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups. In real dimension three, where a result of Hempel ensures that the fundamental group is always residually finite, we then provide a Price type inequality proof of a well-known result of Lott and Lueck. Finally, we give several classes of higher graph manifolds which do indeed have residually finite fundamental groups.

Paper Structure

This paper contains 11 sections, 32 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Extended Graph Manifolds and Pure Complex-Hyperbolic Higher Graph Manifolds
  3. Dimension Estimates
  4. Cheeger-Gromov-Fukaya Theory and Price Inequalities: Proofs of Theorems \ref{['real']} and \ref{['complex']}
  5. Higher Graph Manifolds Without Pure Pieces
  6. Residual Finiteness of Higher Graph Manifolds
  7. Wise's Virtually Special Machinery and Pure Arithmetic Real-Hyperbolic Higher Graph Manifolds
  8. Hempel's Criterion and the Cusp Covering Conjecture
  9. Iterated Partial Doubles
  10. Explicit Examples
  11. The Singer Conjecture in Dimension Three Revisited

Key Result

Theorem 3

Let $M$ be an extended graph manifold with $k\geq 1$ real-hyperbolic pieces $\{(V_j, g_{\mathbb{R}})\}^k_{j=1}$ and residually finite fundamental group. If $\dim_{\mathbb{R}}(M)=2n$, we have Finally, if $\dim_{\mathbb{R}}(M)=2n+1$ we have

Figures (3)

  • Figure 1: Two copies of $S$ with opposite orientations. The red boundary components are $S^1$-bundles with Euler number 3 while the blue ones are $S^1$ bundles with Euler number 1. Adjacent boundary components are glued together via the identity map to produce $M_1$. The symbol "+" refers to the canonical orientation induced by the complex structure, and naturally the symbol "--" refers to the opposite orientation.
  • Figure 2: $4$ copies of $X$ with orientations as shown. Adjacent boundary components are glued together via the identity map to produce $M_2$.
  • Figure 3: $2$ copies of $X$ with opposite orientations. Boundary components are glued together via the restriction of $\bar{\psi}$ according to the given matching to produce $M_3$.

Theorems & Definitions (66)

  • Conjecture 2: Singer Conjecture
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Theorem 8
  • Definition 9
  • Definition 10: Section 2 in Hempel, Definitions 2.12 & 2.13 in LafontBook
  • Remark 11
  • Corollary 12
  • Definition 13
  • ...and 56 more