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Thick embeddings into the Heisenberg group and coarse wirings into groups with polynomial growth

Or Kalifa

Abstract

We bound the volume of thick embeddings of finite graphs into the Heisenberg group, as well as the volume of coarse wirings of finite graphs into groups with polynomial growth. This work follows the work of Kolmogorov-Brazdin, Gromov-Guth and Barret-Hume on thick embeddings of graphs (or complexes) into various spaces. We present here a conjecture of Itai Benjamini that suggest that the lower bound of the volume of thick embeddings of finite graphs into locally finite, non-planar, transitive graphs, obtained by the separation profile, is tight. Let $Y$ be a Cayley graph of a group with polynomial growth, we prove that any finite bounded-degree graph $G$ admits a coarse $C\log(1+|G|)$-wiring into $Y$ with the optimal volume suggested by the conjecture. Additionally, for the concrete case where $Y$ is a Cayley graph of the 3 dimensional discrete Heisenberg group, we prove that any finite bounded-degree graph $G$ admits a $1$-thick embedding into $Y$, with optimal volume up to factor $\log^2(1+|G|)$.

Thick embeddings into the Heisenberg group and coarse wirings into groups with polynomial growth

Abstract

We bound the volume of thick embeddings of finite graphs into the Heisenberg group, as well as the volume of coarse wirings of finite graphs into groups with polynomial growth. This work follows the work of Kolmogorov-Brazdin, Gromov-Guth and Barret-Hume on thick embeddings of graphs (or complexes) into various spaces. We present here a conjecture of Itai Benjamini that suggest that the lower bound of the volume of thick embeddings of finite graphs into locally finite, non-planar, transitive graphs, obtained by the separation profile, is tight. Let be a Cayley graph of a group with polynomial growth, we prove that any finite bounded-degree graph admits a coarse -wiring into with the optimal volume suggested by the conjecture. Additionally, for the concrete case where is a Cayley graph of the 3 dimensional discrete Heisenberg group, we prove that any finite bounded-degree graph admits a -thick embedding into , with optimal volume up to factor .

Paper Structure

This paper contains 27 sections, 19 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Combinatorial Embedding
  3. Proof Overview
  4. Theorem \ref{['thm:A']}
  5. Theorem \ref{['thm:B']}
  6. Paper overview
  7. Definitions
  8. Notations
  9. Cayley Graph
  10. Walk on a Cayley Graph
  11. Random wiring into groups with polynomial growth
  12. Random combinatorial wiring
  13. proof of Theorem \ref{['thm: Random combinatorial wiring']}
  14. Heisenberg group
  15. Coarse to fine in the Heisenberg group
  16. ...and 12 more sections

Key Result

Theorem A

Let $d \in \mathbb{N}$ and let $\Gamma$ be a group with polynomial growth of order $\alpha > 1$. Let $S \subseteq \Gamma$ be a finite generating set. There exists a constant $C = C(d, \Gamma, S)$ such that for any graph $G$ with $n$ vertices and maximum degree $\leq d$, there is a coarse $C\log(1+n)

Figures (1)

  • Figure 1: Small portion of the graph $Cay(\mathbb{H},\{x,y,z\})$. The black edges are labeled $x$, the blue edges are labeled $y$ and the orange edges are labeled $z$.

Theorems & Definitions (38)

  • Definition 1.1
  • Conjecture
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Definition 1.3: Combinatorial Wiring
  • Definition 1.4: combinatorial embedding
  • Theorem 1.1: Random combinatorial wiring into group with polynomial growth
  • Proposition 1.2: Burden relief
  • Proposition 1.3: Removing $z$
  • ...and 28 more