PreGarside monoids and groups, parabolicity, amalgamation, and FC property

TL;DR

This paper introduces preGarside groups, a broader class than Garside groups, studies their properties, and shows they are closed under amalgamation along parabolic subgroups, especially focusing on FC type groups.

Contribution

It defines preGarside groups, characterizes parabolic subgroups, and proves closure under amalgamation, expanding the understanding of Garside-like structures.

Findings

PreGarside groups include all Artin-Tits groups.

Amalgamation along parabolic subgroups preserves preGarside structure.

FC type preGarside groups are closed under amalgamation.

Abstract

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.