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

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Abstract

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 small values of n and g, where g is the genus of the surface.

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Notes

  1. “Schwarz” is a transliteration from Russian; alternative spellings “Švarc”, “Schwartz” sometimes appear in the literature.

  2. It is curious to note that the same is not true for the action of \(\mathrm {Homeo}(M)\) as was recently proven [42].

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Acknowledgements

MB was partially supported by Humboldt Research Fellowship. MM was partially supported by NCN (Sonatina 2018/28/C/ST1/00542) and by the GAČR project 19-05271Y and by RVO: 67985840. ES was partially supported by an NSERC Discovery Grant, by the Fondation Courtois, and by a Sloan Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University, 84105, Be’er Sheva, Israel

    Michael Brandenbursky

  2. University of Wrocław, Faculty of Mathematics, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Michał Marcinkowski

  3. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Michał Marcinkowski

  4. Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville, Montreal, QC, H3C 3J7, Canada

    Egor Shelukhin

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  1. Michael Brandenbursky
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Correspondence to Egor Shelukhin.

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Brandenbursky, M., Marcinkowski, M. & Shelukhin, E. The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms. Sel. Math. New Ser. 28, 74 (2022). https://doi.org/10.1007/s00029-022-00784-0

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