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A computable version of Hall's Harem Theorem and Geometric von Neumann Conjecture

Karol Duda

TL;DR

The paper delivers computable versions of Hall's harem theory and Schneider–Whyte results in the setting of non-amenable coarse spaces by constructing a computable perfect (1,d−1)-matching that encodes a computable (d−1)-to-1 function with tightly controlled cycles. The core method is a priority-style, stepwise construction on highly computable, fully reflected bipartite graphs that preserves the computable Hall d-harem property while forcing cycle-length bounds. The main contributions are a computable version of Hall's harem theorem with cycles, a computable analogue of Schneider's geometric non-amenability results (including a computable d-regular forest and decidable component connectivity), and the consequent computable corollaries for wobbling groups and the geometric von Neumann conjecture. Together these results advance the effective understanding of amenability phenomena and provide algorithmic tools for analyzing the large-scale geometry of non-amenable spaces.

Abstract

We prove a computable version of the Hall Harem Theorem where the matching realizes a unary function with controlled sizes of cycles. We apply it to non-amenable computable coarse spaces. As a result, we obtain a computable version of the geometric von Neumann conjecture.

A computable version of Hall's Harem Theorem and Geometric von Neumann Conjecture

TL;DR

The paper delivers computable versions of Hall's harem theory and Schneider–Whyte results in the setting of non-amenable coarse spaces by constructing a computable perfect (1,d−1)-matching that encodes a computable (d−1)-to-1 function with tightly controlled cycles. The core method is a priority-style, stepwise construction on highly computable, fully reflected bipartite graphs that preserves the computable Hall d-harem property while forcing cycle-length bounds. The main contributions are a computable version of Hall's harem theorem with cycles, a computable analogue of Schneider's geometric non-amenability results (including a computable d-regular forest and decidable component connectivity), and the consequent computable corollaries for wobbling groups and the geometric von Neumann conjecture. Together these results advance the effective understanding of amenability phenomena and provide algorithmic tools for analyzing the large-scale geometry of non-amenable spaces.

Abstract

We prove a computable version of the Hall Harem Theorem where the matching realizes a unary function with controlled sizes of cycles. We apply it to non-amenable computable coarse spaces. As a result, we obtain a computable version of the geometric von Neumann conjecture.

Paper Structure

This paper contains 31 sections, 19 theorems, 87 equations, 16 figures.

Table of Contents

  1. Introduction
  2. A computable version of Scheider's theorem
  3. Hall's harem theorem
  4. Preliminaries
  5. General preliminaries
  6. Reflections
  7. Controlled sizes of cycles.
  8. Computable versions of Hall's d-harem theorem.
  9. A computable version of Hall's harem Theorem with cycles, preliminaries
  10. Main Theorem and the aim of the construction
  11. Notation used in the construction
  12. Construction
  13. Step 1, part 1
  14. Remarks before part 2
  15. Step 1, part 2
  16. ...and 16 more sections

Key Result

Theorem 1.1

A uniformly discrete metric space of uniformly bounded geometry is non-amenable if and only if it admits a partition whose pieces are uniformly Lipschitz embbeded copies of the $4$-regular tree.

Figures (16)

  • Figure 1: Single step.
  • Figure 2: The first part of step 1, the $\mathfrak{M}^1_{0}$-fan of $u_0$ where $M^0_0$ is green.
  • Figure 3: Step 1, part 3. $\mathfrak{M}^2_{0}$ is red and $v_{u_0}$ is matched with $u$. We want the edge $(u^{1}_0, v_{u_0})$ to be in $M_0$. Force the situation from Figure 4.
  • Figure 4: $M^1_{0}$ is green. It is possible that the purple fan consisting of edges $(\dot{u}^{\perp}_0,\dot{v}^{\perp}_{0,1}),(\dot{u}^{\perp}_0,\dot{v}^{\perp}_{0,2})$ will be added to $\Gamma^{(1)\perp}$.
  • Figure 5: Detailed version of the algorithm used at part 1 of step $n+1$.
  • ...and 11 more figures

Theorems & Definitions (68)

  • Theorem 1.1: Geometric Von Neumann Conjecture, Whyte
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem A
  • Theorem B
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • ...and 58 more