Resumen de Admissible submonoids of Artin-Tits monoids

Anatole Castella

• We show the analogue of Mühlherr�s [B. Mühlherr, Coxeter groups in Coxeter groups, in: Finite Geometry and Combinatorics, Cambridge University Press, 1993, pp. 277�287] for Artin�Tits monoids and for Artin�Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin�Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin�Tits monoid (resp. group).

  This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin�Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [J. Crisp, Injective maps between Artin groups, in: Geom. Group Theory Down Under (Canberra 1996), de Gruyter, Berlin, 1999, pp. 119�137] and generalized by Godelle in [E. Godelle, Morphismes injectifs entre groupes d�Artin-Tits, Algebr. Geom. Topol. 2 (2002) 519�536].

  We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Mühlherr in [B. Mühlherr, Some contributions to the theory of buildings based on the gate property, Dissertation, Tübingen, 1994]. This leads us to the classification of Crisp�s LCM-homomorphisms.