Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups

  • Michael Brandenbursky

    Ben Gurion University of the Negev, Beer Sheva, Israel
  • Michał Marcinkowski

    Ben Gurion University of the Negev, Beer Sheva, Israel, Regensburg University, Germany, and Wroclaw University, Poland
Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups cover
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Abstract

Let Fn\mathbf{F}_n be the free group on nn generators and Γg\boldsymbol{\Gamma}_g the surface group of genus gg. We consider two particular generating sets: the set of all primitive elements in Fn\mathbf{F}_n and the set of all simple loops in Γg\boldsymbol{\Gamma}_g. We give a complete characterization of distorted and undistorted elements in the corresponding Aut-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on F2\mathbf{F}_2 that are Aut(F2)(\mathbf{F}_2)-invariant. This answers an open problem posed by Miklós Abért.

Cite this article

Michael Brandenbursky, Michał Marcinkowski, Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups. Comment. Math. Helv. 94 (2019), no. 4, pp. 661–687

DOI 10.4171/CMH/470