# Ordering Garside groups

## TL;DR

This paper introduces Dehornoy structures for Garside groups, providing conditions for their existence, and applies these to show certain Artin groups admit such structures, linking them to known orders.

## Contribution

It defines Dehornoy structures on Garside groups, establishes criteria for their existence, and demonstrates their application to specific Artin groups, connecting to existing orderings.

## Key findings

- Artin groups of type A and I2(m) admit Dehornoy structures
- Dehornoy orders coincide with known Artin group orders
- Conditions for Garside groups to have Dehornoy structures established

## Abstract

We introduce a condition on Garside groups that we call Dehornoy structure. An iteration of such a structure leads to a left order on the group. We show conditions for a Garside group to admit a Dehornoy structure, and we apply these criteria to prove that the Artin groups of type A and I 2 (m), m $\ge$ 4, have Dehornoy structures. We show that the left orders on the Artin groups of type A obtained from their Dehornoy structures are the Dehornoy orders. In the case of the Artin groups of type I 2 (m), m $\ge$ 4, we show that the left orders derived from their Dehornoy structures coincide with the orders obtained from embeddings of the groups into braid groups. 20F36