Otdfbt

Suppose $G$ is a group. Let’s call $a, b in G$ commensurable, iff $exists m, n in mathbbZ setminus 0$, such that $a^n = b^m$. One can see, that commensurability is an equivalence relationship:
$$a = a$$
$$(a^n = b^m) to (b^m = a^n)$$
$$((a^n = b^m) cap (b^p = c^q)) to (a^np = c^mq)$$

Thus we can divide the elements of a group into commensurability classes.

All finite-order elements of a group form a commensurability class

If $a$ and $b$ are finite-order elements, then we can take $n$ and $m$ as their respective orders.

If $a^k = e$ and $a^n = b^m$, $b^mk = e$.

My question is:

What is the possible structure of commensurability classes of infinite-order elements?

What do I know about them:

Suppose, $a$ and $b$ are commensurable commuting infinite-order elements of $G$. Then $exists c in G$, such that $a, b in langle c rangle$

Suppose $a^n = b^m$. Then $c = a^q b^p$, where $p$ and $q$ are integers, such that $pn + qm = gcd(n, m)$.

Now an important question to which I do not know an answer is:

Does there exist a group that contains a pair of commensurable non-commuting infinite-order elements?

The negative answer to this question implies that all commensurability classes of infinite order elements are of the form $Q setminus e$, where $Q$ is a maximal locally cyclic subgroup of $G$. However, I do not know, how to prove that the answer is negative (or does such group actually exist?)

Consider the group $S(mathbb N)$ of all bijections of the set of natural numbers. Now consider the elements $(2,4,1,3)sigma$ and $(1,2,3,4)sigma$, where $sigma$ is any infinite-order bijection of $5,6,...$.

There are plenty of examples. For instance, for any $n,mge 2$, in the group with presentation $langle x,y|x^m=y^nrangle$, the elements $x,y$ have infinite order, do not commute, and are commensurate (I prefer "commensurate" to "commensurable" since there's no choice). (For $n=m=2$ this is the $pi_1$ of the Klein bottle). Idem, $x,y$ are commensurate in the Baumslag-Solitar group $langle t,x|tx^nt^-1=x^mrangle$ for $y=txt^-1$.

The first question "What is the possible structure of" seems too imprecise for an answer. Anyway there are many natural things one can do with this. For instance, that a class of some infinite order element is invariant under conjugation is an interesting property; this is called a commensurated infinite cyclic subgroup.