Parabolic subgroups of Garside groups

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains all Artin–Tits groups of spherical type. We show that the well-known notion of a parabolic subgroup of an Artin–Tits group can be extended to the framework of Garside groups so that most of the standard properties known in the Artin–Tits groups case are preserved. The extension is not trivial and it requires a new approach. We also define the more general notion of a Garside subgroup of a Garside group that nicely extends the notion of an LCM-homomorphism between Artin–Tits groups.