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

Michael Brandenbursky, Michał Marcinkowski

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Let Fn be the free group on n generators and Γg the surface group of genus g. We consider two particular generating sets: the set of all primitive elements in Fn and the set of all simple loops in Γg. 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 that are Aut(F2)-invariant. This answers an open problem posed by Miklós Abért.

Original languageEnglish
Pages (from-to)661-687
Number of pages27
JournalCommentarii Mathematici Helvetici
Issue number4
StatePublished - 1 Jan 2019


  • Free groups
  • Invariant norms
  • Mapping class groups
  • Quasi-morphisms

ASJC Scopus subject areas

  • General Mathematics


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