The Artin monoid Cayley graph

Abstract

In this paper we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group $A_\Gamma$ with respect to the (infinite) generating set given by the associated Artin monoid $A^+_\Gamma$. In a previous paper, the first three authors introduced a monoid Deligne complex and showed that this complex is contractible for all Artin groups. In this paper, we show that the Artin monoid Cayley graph is quasi-isometric to a modification of the Deligne complex for $A_\Gamma$ obtained by coning off translates of the monoid Deligne complex. We then address the question of when the monoid Cayley graph has infinite diameter. We conjecture that this holds for all Artin groups of infinite type. We give a set of criteria that imply infinite diameter, and using existing solutions to the word problem for large-type Artin groups and 3-free Artin groups, we prove that the conjecture holds for any Artin group containing a 3-generator subgroup of one of these two types.