Powers of Dehn twists generating right-angled Artin groups

TL;DR

This paper establishes a specific bound on the powers of Dehn twists needed to generate right-angled Artin groups from collections of curves on surfaces, extending previous results and providing universal bounds in some cases.

Contribution

It provides a concrete exponent bound for powers of Dehn twists to generate right-angled Artin groups, improving upon earlier bounds that depended on hyperbolic structures.

Findings

Bound for powers of Dehn twists depending on intersection numbers

Extension of Koberda's previous results with explicit bounds

Universal bounds depending only on surface topology in certain cases

Abstract

We give a bound for the exponents of powers of Dehn twists to generate a right-angled Artin group. Precisely, if $\mathcal{F}$ is a finite collection of pairwise distinct simple closed curves on a finite type surface and if $N$ denotes the maximum of the intersection numbers of all pairs of curves in $\mathcal{F}$, then we prove that $\{T_\gamma^n \,\vert\, \gamma \in \mathcal{F} \}$ 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.