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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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