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

• Michael Brandenbursky ^[3] ; Michał Marcinkowski ^[1] ; Egor Shelukhin ^[2]
  1. [1] University of Wrocław

     University of Wrocław

     Breslavia, Polonia

     
  2. [2] University of Montreal

     University of Montreal

     Canadá

     
  3. [3] Ben Gurion University, Israel

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 4, 2022
• Idioma: inglés
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• Resumen

  • 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 areapreserving diffeomorphisms endowed with the L p-metric, and that all GambaudoGhys 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 small values of n and g, where g is the genus of the surface.