# Commutator Width of a direct limit of hyperbolic groups

Is it known if the direct limit of hyperbolic groups can have finite commutator width? Every hyperbolic group has infinite verbal width for any word $w$, so in particular for the commutator word $w=x^{-1}y^{-1}xy$ (http://arxiv.org/abs/1107.3719)

Especially I would like to know if the examples of infinite torsion groups with exactly $n$ conjugacy classes as constructed by Ol'shanksii in the book "Geometry of Defining Relations of Groups" necessarily have to have infinite commutator width.

More generally I am interested in examples of infinite groups having only finitely many conjugacy classes.

• I do not understand. Any group having only finitely many conjugacy classes must have finite commutator width, because conjugate elements have the same commutator length. – Anton Klyachko Dec 12 '13 at 18:08
• Anton, yes you're right, for finitely many conjugacy classes this is easy to see. I realised that too late. – Elisabeth Fink Dec 13 '13 at 19:30

Assuming that you refer to Ivanov's construction from Ol'shanskii's book of a $2$-generated infinite group $G$ of exponent $p$, for a large prime $p$, having exactly $p$ conjugacy classes, then the commutator width in this group is bounded above by $p-1$. The reason for this is that for any non-trivial commutator $w$ in $G$, every element of $G$ is conjugate to $w^k$ for some $0 \le k \le p-1$ (because, by construction, $\langle w \rangle$ is a subgroup of order $p$ and intersects each conjugacy class of $G$). It is known, of course, that $G$ is a direct limit of hyperbolic groups.