# Hyperbolic structures for Artin-Tits groups of spherical type

## TL;DR

This paper explores various algebraic hyperbolic structures on irreducible Artin-Tits groups of spherical type, drawing parallels with known geometric structures on braid groups to deepen understanding of their properties.

## Contribution

It introduces several algebraic hyperbolic structures for Artin-Tits groups of spherical type and clarifies their interrelations, expanding the theoretical framework of these groups.

## Key findings

- Identification of multiple algebraic hyperbolic structures
- Connections established between algebraic and geometric hyperbolic structures
- Enhanced understanding of the relations among different hyperbolic models

## Abstract

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natural actions of these groups on curve graphs and (modified) arc graphs of punctured disks.