Volume 13, issue 2 (2013)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 D2 be the open unit disc in the Euclidean plane and let G := Diff(D2,area) be the group of smooth compactly supported area-preserving diffeomorphisms of D2. For every natural number k we construct an injective homomorphism Zk G, which is bi-Lipschitz with respect to the word metric on Zk 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