Distortion of imbeddings of groups of intermediate growth into metric spaces

TL;DR

The paper constructs groups of subexponential growth that exhibit arbitrarily large distortion when embedded into certain metric spaces, including Banach spaces like Hilbert space, highlighting limitations of embeddings.

Contribution

It introduces a method to produce groups of subexponential growth with embeddings into metric spaces that have arbitrarily large distortion, extending understanding of embedding limitations.

Findings

Groups of subexponential growth can have embeddings with arbitrarily large distortion.

Applicable to B-convex Banach spaces, including Hilbert spaces.

Demonstrates limitations of coarse embeddings for certain groups.

Abstract

For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $\rho$, we produce a group of subexponential word growth all of whose imbeddings in $\mathcal X$ have distortion worse than $\rho$. This applies in particular to any B-convex Banach space $\mathcal X$, such as Hilbert space.