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

Michael Brandenbursky; Jarek Kedra

doi:10.2140/agt.2013.13.795

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

Michael Brandenbursky, Jarek Kedra

Mathematical Science

Research output: Contribution to journal › Article

13 Citations (Scopus)

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.

Original language	English
Pages (from-to)	795-816
Number of pages	23
Journal	Algebraic & Geometric Topology
Volume	13
Issue number	2
DOIs	https://doi.org/10.2140/agt.2013.13.795
Publication status	Published - 28 Mar 2013

Keywords

area-preserving diffeomorphisms
braid groups
quasimorphisms
quasi-isometric embeddings
bi-invariant metrics

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Brandenbursky, M., & Kedra, J. (2013). On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc. Algebraic & Geometric Topology, 13(2), 795-816. https://doi.org/10.2140/agt.2013.13.795

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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",

author = "Michael Brandenbursky and Jarek Kedra",

note = "Both authors would like to thank the anonymous referee for careful reading of our paper and for his/her helpful comments and remarks. This work has been done during the first author stay in Aberdeen and in Mathematisches Forschungsinstitut Oberwolfach. The first author wishes to express his gratitude to both institutes. He was supported by ESF grant number 4824 and by the Oberwolfach Leibniz fellowship. The visit in Aberdeen was supported by the CAST network.",

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AB - 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.

KW - area-preserving diffeomorphisms

KW - braid groups

KW - quasimorphisms

KW - quasi-isometric embeddings

KW - bi-invariant metrics

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