  We define the notion of preGarside group slightly lightening the definition
of Garside group so that all Artin-Tits groups are preGarside groups. This
paper intends to give a first basic study on these groups. Firstly, we
introduce the notion of parabolic subgroup, we prove that any preGarside group
has a (partial) complemented presentation, and we characterize the parbolic
subgroups in terms of these presentations. Afterwards we prove that the
amalgamated product of two preGarside groups along a common parabolic subgroup
is again a preGarside group. This enables us to define the family of preGarside
groups of FC type as the smallest family of preGarside groups that contains the
Garside groups and that is closed by amalgamation along parabolic subgroups.
Finally, we make an algebraic and combinatorial study on FC type preGarside
groups and their parabolic subgroups.
