Cyclic amalgams, HNN extensions, and Garside one-relator groups

TL;DR

This paper explores the structure of Garside groups, showing their stability under certain free products and HNN extensions, and characterizing when non-cyclic one-relator groups are Garside based on their centers.

Contribution

It demonstrates that Garside groups are closed under specific amalgamated free products and cyclic HNN extensions, and characterizes Garside non-cyclic one-relator groups via their centers.

Findings

Garside groups are closed under some free products with cyclic amalgamation.

Tree products of infinite cyclic groups are Garside groups.

Non-cyclic one-relator groups are Garside iff their center is nontrivial.

Abstract

Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.