Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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].
References
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Bibliographic Information
  • Xiaolei Wu
  • Affiliation: Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No.2005 Songhu Road, Shanghai 200438, People’s Republic of China
  • MR Author ID: 1071753
  • ORCID: 0000-0003-2064-4455
  • Email: xiaoleiwu@fudan.edu.cn
  • Shengkui Ye
  • Affiliation: NYU Shanghai, No.567 Yangsi West Rd, Pudong New Area, Shanghai 200124, People’s Republic of China; and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, People’s Republic of China
  • MR Author ID: 834051
  • Email: sy55@nyu.edu
  • Received by editor(s): January 20, 2024
  • Received by editor(s) in revised form: October 28, 2024
  • Published electronically: April 4, 2025
  • Additional Notes: The first author is currently a member of LMNS and was supported by a starter grant at Fudan University. The second author was supported by NSFC (No. 11971389).
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4707-4737
  • MSC (2020): Primary 20F65
  • DOI: https://doi.org/10.1090/tran/9408