2012/04/25 by Eddy Godelle, Luis Paris, Godelle, Eddy +1
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #math.GR
paper · pdf · doi:10.48550/arxiv.1204.5672
arxiv created 2012/04/25 · openalex publication_date 2012/04/25 · arxiv updated 2012/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.