# Generalising right-angled Artin groups

An Artin group $G$ is determined by its Coxeter matrix $M$. This is a symmetric $n \times n$ matrix with entries from $\lbrace 2, 3, \ldots, \infty \rbrace$ that determine the relations between the generators of the group. If all of the entries are in $\lbrace 2, \infty \rbrace$ then we say that $G$ is a right-angled Artin group''.

Is there a name for an Artin group in which all of the entries of its Coxeter matrix are in $\lbrace 2, 3, \infty \rbrace$ (or alternatively $\lbrace 2, 3 \rbrace$)?

Note that, for example, all braid groups are of this type.

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## 1 Answer

These groups are called Artin groups of small type. See, for example, Crisp, John; Paris, Luis, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math. 145 (2001), no. 1, 19–36.

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Thanks, small type refers to the {2,3} case, correct? Is there a similar name for the {2,3,\infty} case? –  Mark Bell Jun 14 '12 at 11:19
I would think that it is about $(2,3,\infty)$ case. But I have not looked at Crisp-Paris for quite some time. I remember that they embedded every Artin group into one with small type. –  Mark Sapir Jun 14 '12 at 11:57
Actually you are right: small type is for exponents $\\{2,3\\}$. –  Mark Sapir Jun 14 '12 at 12:31