Strong distortion in transformation groups

TL;DR

This paper investigates the properties of transformation groups, demonstrating strong distortion and boundedness in certain diffeomorphism groups, and addresses a longstanding problem related to embedding properties in these groups.

Contribution

It proves strong distortion in various diffeomorphism groups and establishes a relative Higman embedding property, solving a problem posed by Schreier.

Findings

Groups $ ext{Diff}^r_0( ext{R}^n)$ and $ ext{Diff}^r( ext{R}^n)$ have strong distortion for specified r.

Every abstract length function on these groups is bounded.

Groups $ ext{Diff}_0^r(M)$ satisfy a relative Higman embedding property for certain manifolds.

Abstract

We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$. This implies in particular that every abstract length function on these groups is bounded. With related techniques we show that, for $M$ a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the groups $\mathrm{Diff}_0^r(M)$ satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book.