Embeddability of generalized wreath products and box spaces

TL;DR

This paper introduces a generalized wreath product construction for metric spaces and establishes conditions under which coarse embeddability into Hilbert space is preserved, also exploring stability of box space embeddability under wreath products.

Contribution

It generalizes wreath product constructions to metric spaces, defines the path lifting property, and analyzes embeddability and stability of box spaces under wreath products.

Findings

Coarse embeddability is preserved under the new wreath product construction.

Bounds on compression of wreath products are derived.

New examples of coarsely embeddable spaces without property A are provided.

Abstract

Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces X,Y,Z and derive a condition, called the (delta-polynomial) path lifting property, such that coarse embeddability of X,Y and Z implies coarse embeddability of X\wr_Z Y. We also give bounds on the compression of X\wr_Z Y in terms of delta and the compressions of X,Y and Z. Next, we investigate the stability of the property of admitting a box space which coarsely embeds into a Hilbert space under the taking of wreath products. We show that if an infinite finitely generated residually finite group H has a coarsely embeddable box space, then G\wr H has a coarsely embeddable box space if G is finitely generated abelian. This leads, in particular, to new examples of bounded geometry coarsely embeddable metric spaces without property A.