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

TL;DR

This paper investigates volume bounds for thick embeddings and coarse wirings of finite graphs into the Heisenberg group and groups with polynomial growth, providing new bounds and confirming conjectures in geometric group theory.

Contribution

It proves new volume bounds for thick embeddings into the Heisenberg group and coarse wirings into polynomial growth groups, confirming a conjecture by Itai Benjamini.

Findings

Bound the volume of thick embeddings into the Heisenberg group.

Establish coarse wiring bounds into groups with polynomial growth.

Confirm conjecture on the tightness of lower bounds for embeddings into non-planar transitive graphs.

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 $\log^2(1+|G|)$.