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arXiv · 1610.06720

Strong distortion in transformation groups

Frédéric Le Roux·Kathryn Mann
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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.

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Frédéric Le Roux, Kathryn Mann. 2017-09-28. Strong distortion in transformation groups. https://doi.org/10.1112/blms.12108

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