# Tits alternative for Artin groups of type FC

## TL;DR

This paper establishes a criterion based on cube stabilisers for groups acting on CAT(0) cube complexes to satisfy the strong Tits alternative, and applies it to prove this property for all Artin groups of type FC.

## Contribution

It introduces a new stabiliser-based criterion for the strong Tits alternative and proves it holds for all Artin groups of type FC.

## Key findings

- All Artin groups of type FC satisfy the strong Tits alternative.
- The criterion simplifies checking the Tits alternative for groups acting on CAT(0) cube complexes.
- The approach relies on properties of cube stabilisers.

## Abstract

Given a group action on a finite-dimensional CAT(0) cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser satisfies the strong Tits alternative. We use it to prove that all Artin groups of type FC satisfy the strong Tits alternative.