Powers of Dehn twists generating right-angled Artin groups

Seo, Donggyun

doi:10.2140/agt.2021.21.1511

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Powers of Dehn twists generating right-angled Artin groups

Donggyun Seo

Algebraic & Geometric Topology 21 (2021) 1511–1533

DOI: 10.2140/agt.2021.21.1511

Abstract

We give a bound for the exponents of powers of Dehn twists to generate a right-angled Artin group. Precisely, if $ℱ$ is a finite collection of pairwise distinct simple closed curves on a surface and if $N$ denotes the maximum of the intersection numbers of all pairs of curves in $ℱ$ , then we prove that ${T_{γ}^{n} ∣ γ \in ℱ}$ generates a right-angled Artin group for all $n \geq N^{2} + N + 3$ . This extends a previous result of Koberda, who proved the existence of a bound possibly depending on the underlying hyperbolic structure of the surface. In the course of the proof, we obtain a universal bound depending only on the topological type of the surface in certain cases, which partially answers a question due to Koberda.

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Keywords

Dehn twist, right-angled Artin group, mapping class group

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E08, 57M60

References

Publication

Received: 10 December 2019

Revised: 29 April 2020

Accepted: 3 June 2020

Published: 11 August 2021

Authors

Donggyun Seo
School of Mathematics Korea Institute for Advanced Study Seoul South Korea

Volume 21, issue 3 (2021)