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

Michael Brandenbursky; Michał Marcinkowski

doi:10.4171/CMH/470

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

Michael Brandenbursky, Michał Marcinkowski

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

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

Original language	English
Pages (from-to)	661-687
Number of pages	27
Journal	Commentarii Mathematici Helvetici
Volume	94
Issue number	4
DOIs	https://doi.org/10.4171/CMH/470
State	Published - 1 Jan 2019

Keywords

Free groups
Invariant norms
Mapping class groups
Quasi-morphisms

ASJC Scopus subject areas

General Mathematics

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10.4171/CMH/470

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title = "Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups",

abstract = "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 Mikl{\'o}s Ab{\'e}rt.",

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AU - Marcinkowski, Michał

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N2 - 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.

AB - 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.

KW - Free groups

KW - Invariant norms

KW - Mapping class groups

KW - Quasi-morphisms

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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