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

Michael Brandenbursky, Jarek Kedra

Research output: Contribution to journalArticle

10 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 languageEnglish
Pages (from-to)795-816
Number of pages23
JournalAlgebraic & Geometric Topology
Volume13
Issue number2
DOIs
Publication statusPublished - 28 Mar 2013

Fingerprint

Diffeomorphisms
Metric
Euclidean plane
Natural number
Injective
Homomorphism
Unit Disk
Lipschitz

Keywords

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

Cite this

On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc. / Brandenbursky, Michael; Kedra, Jarek.

In: Algebraic & Geometric Topology, Vol. 13, No. 2, 28.03.2013, p. 795-816.

Research output: Contribution to journalArticle

@article{363794252ee44c2bb18509cac8478e36,
title = "On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc",
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.",
year = "2013",
month = "3",
day = "28",
doi = "10.2140/agt.2013.13.795",
language = "English",
volume = "13",
pages = "795--816",
journal = "Algebraic & Geometric Topology",
issn = "1472-2747",
publisher = "Agriculture.gr",
number = "2",

}

TY - JOUR

T1 - On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc

AU - Brandenbursky, Michael

AU - Kedra, Jarek

N1 - 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.

PY - 2013/3/28

Y1 - 2013/3/28

N2 - 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.

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

U2 - 10.2140/agt.2013.13.795

DO - 10.2140/agt.2013.13.795

M3 - Article

VL - 13

SP - 795

EP - 816

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

IS - 2

ER -