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arXiv · 2312.08681

Splittings and poly-freeness of triangle Artin groups

Xiaolei Wu·Shengkui Ye
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Abstract

We prove that the triangle Artin group $\mathrm{Art}_{23M}$ splits as a graph of free groups if and only if $M$ is greater than $5$ and even. This answers two questions of Jankiewicz \cite[Question 2.2, Question 2.3]{Jan21} in the negative. Combined with the results of Squier and Jankiewicz, this completely determines when a triangle Artin group splits as a graph of free groups. Furthermore, we prove that the triangle Artin groups are virtually poly-free when the labels are not of the form $(2,3, 2k+1)$ with $k\geq 3$. This partially answers a question of Bestvina \cite{Be99}.

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Xiaolei Wu, Shengkui Ye. 2023-12-14. Splittings and poly-freeness of triangle Artin groups. https://arxiv.org/abs/2312.08681

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