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Convexity of parabolic subgroups in Artin groups. (English) Zbl 1308.20037
From the introduction: We establish the following theorem. Theorem 1.2. Let \(\Gamma\) be a Coxeter graph with vertex set \(S\), let \((A,\Sigma)\) be the corresponding Artin system, and let \(T\) be a subset of \(S\). Then \(A_T\) is a convex subgroup of \(A\) with respect to \(\Sigma\).
Although Theorem 1.2 may seem natural, it is a surprise for the experts. It follows from prior work of D. F. Holt and S. Rees [Proc. Lond. Math. Soc. (3) 104, No. 3, 486-512 (2012; Zbl 1275.20034); Groups Complex. Cryptol. 5, No. 1, 1-23 (2013; Zbl 1284.20036)] that the statement holds for Artin groups of large type (those with all \(m_{s,t}\geq 3\)) and a slightly more general class, ‘sufficiently large’ Artin groups. But as far as we know, it was not even known for the braid group \(\mathcal B_m\) embedded in \(\mathcal B_n\), although the proof in this case is easy (see Proposition 2.1). Theorem 1.2 also comes as a surprise because, in general, the family of standard generators (that is, \(\Sigma\)) is not the best for studying combinatorial questions on Artin groups. In the most well-understood case, when \(A\) is of spherical type, a larger generating set is generally used. This is called the set of simple elements and we denote it by \(\mathcal S\). It follows from [R. Charney, Math. Ann. 301, No. 2, 307-324 (1995; Zbl 0813.20042)] that \((A_T,\mathcal S_T)\) is isometrically embedded in \((A,\mathcal S)\), but the image is not convex since there are \(\mathcal S\)-geodesics for elements of \(A_T\) whose terms do not belong to \(\mathcal S_T\sqcup\mathcal S_T^{-1}\).

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20F65 Geometric group theory
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