Volume 13, issue 2 (2013)

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On the autonomous metric on the group of area-preserving diffeomorphisms of the $2$–disc

Michael Brandenbursky and Jarek Kędra

Algebraic & Geometric Topology 13 (2013) 795–816
Abstract

Let ${D}^{2}$ be the open unit disc in the Euclidean plane and let $G:=Diff\left({D}^{2},area\right)$ be the group of smooth compactly supported area-preserving diffeomorphisms of ${D}^{2}$. For every natural number $k$ we construct an injective homomorphism ${Z}^{k}\to G$, which is bi-Lipschitz with respect to the word metric on ${Z}^{k}$ and the autonomous metric on $G$. We also show that the space of homogeneous quasimorphisms vanishing on all autonomous diffeomorphisms in the above group is infinite-dimensional.

Keywords
area-preserving diffeomorphisms, braid groups, quasimorphisms, quasi-isometric embeddings, bi-invariant metrics
Primary: 57S05