# 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.

In fact, the above argument together with a different construction of the group you are looking for, is given in Alexey Muranov's paper "Diagrams with Selection and Method for Constructing Boundedly Generated and Boundedly Simple Groups" (Comm. Algebra 33 (2005), no. 4), http://arxiv.org/abs/math/0404472.

As for the last question, infinitely generated examples of infinite groups with finitely many conjugacy classes are constructed in the book of Lyndon and Schupp. The first finitely generated infinite groups with two conjugacy classes were produced by Denis Osin in "Small cancellations over relatively hyperbolic groups and embedding theorems." (Ann. of Math. (2) 172 (2010), no. 1, 1–39). If I remember correctly, in this paper Denis points out that the latter groups cannot be limits of word hyperbolic groups.