Splittings and poly-freeness of triangle Artin groups

Wu, Xiaolei; Ye, Shengkui

doi:10.1090/tran/9408

Splittings and poly-freeness of triangle Artin groups
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by Xiaolei Wu and Shengkui Ye;

Trans. Amer. Math. Soc. 378 (2025), 4707-4737

DOI: https://doi.org/10.1090/tran/9408

Published electronically: April 4, 2025

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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 [Groups Geom. Dyn. 18 (2024), pp. 91–108; Question 2.2, Question 2.3] 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 [Geom. Topol. 3 (1999), pp. 269–302].

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Splittings and poly-freeness of triangle Artin groups
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Abstract: