Parabolic subgroups inside parabolic subgroups of Artin groups

Blufstein, Martín; Paris, Luis

doi:10.1090/proc/16289

Parabolic subgroups inside parabolic subgroups of Artin groups
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by Martín A. Blufstein and Luis Paris

Proc. Amer. Math. Soc. 151 (2023), 1519-1526

DOI: https://doi.org/10.1090/proc/16289

Published electronically: January 26, 2023

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Abstract:

We prove that a parabolic subgroup $P$ contained in another parabolic subgroup $P’$ of an Artin group $A$ is a parabolic subgroup of $P’$. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction $A \to P$ of the inclusion map from a parabolic subgroup $P$ into $A$. This retraction was implicitly constructed in a previous paper by Charney and the second author [Bull. Lond. Math. Soc. 46 (2014), pp. 1248–1255].

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