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

Michael Brandenbursky; Michał Marcinkowski

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arXiv · 1702.01662

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

Michael Brandenbursky·Michał Marcinkowski

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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 $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklós Abért.

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BibTeX RIS

Michael Brandenbursky, Michał Marcinkowski. 2018-09-11. Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups. https://arxiv.org/abs/1702.01662

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