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

Brandenbursky, Michael; Kędra, Jarek

doi:10.2140/agt.2013.13.795

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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

DOI: 10.2140/agt.2013.13.795

Abstract

Let $D^{2}$ be the open unit disc in the Euclidean plane and let $G : = Diff (D^{2}, area)$ 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

Mathematical Subject Classification 2010

Primary: 57S05

References

Publication

Received: 20 July 2012

Revised: 18 September 2012

Accepted: 11 November 2012

Published: 28 March 2013

Authors

Michael Brandenbursky
Department of Mathematics Vanderbilt University 1326 Stevenson Center Nashville, TN 37240 USA

Jarek Kędra
Institute of Mathematics University of Aberdeen Aberdeen AB24 3UE UK

Volume 13, issue 2 (2013)