Compression bounds for wreath products

TL;DR

This paper proves that the Hilbert compression exponent remains positive for wreath products of groups with positive exponents, and provides bounds for specific cases, advancing understanding of geometric group embeddings.

Contribution

It establishes that positive Hilbert compression exponents are preserved under wreath product constructions, including coarse embeddings, and answers a specific open question.

Findings

Hilbert compression exponent is positive for wreath products of groups with positive exponents

The exponent for Z wr (Z wr Z) is at least 1/4

Results extend understanding of embeddings of complex group structures

Abstract

We show that if $G$ and $H$ are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath $G \wr H$. We also prove an analogous result for coarse embeddings of wreath products. In the special case $G=\Z$, $H=\Z \wr \Z$ our result implies that the Hilbert compression exponent of $\Z\wr (\Z\wr\Z)$ is at least 1/4, answering a question posed as to whether it has positive Hilbert compression exponent.