2
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

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.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thanks, small type refers to the {2,3} case, correct? Is there a similar name for the {2,3,\infty} case? $\endgroup$
    – Mark Bell
    Commented Jun 14, 2012 at 11:19
  • $\begingroup$ 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. $\endgroup$
    – user6976
    Commented Jun 14, 2012 at 11:57
  • $\begingroup$ Actually you are right: small type is for exponents $\\{2,3\\}$. $\endgroup$
    – user6976
    Commented Jun 14, 2012 at 12:31

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .