The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

Michael Brandenbursky, Michał Marcinkowski, Egor Shelukhin

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We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of $n$ points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the $L^p$-metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on $n$ strands are Lipschitz. This was conjectured to hold, yet proven only for low values of $n$ and the genus $g$ of the surface.
Original languageEnglish GB
JournalarXiv preprint
StatePublished - 1 May 2021


  • Mathematics - Geometric Topology
  • Mathematics - Differential Geometry
  • Mathematics - Group Theory
  • Mathematics - Symplectic Geometry


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