Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups

Abstract:

It is easy to pose questions about the free lattice-ordered group ${F_\eta }$ of rank $\eta > 1$ whose answers$^{2}$ are "obvious", but difficult to verify. For example: 1. What is the center of ${F_\eta }$? 2. Is ${F_\eta }$ directly indecomposable? 3. Does ${F_\eta }$ have a basic element? 4. Is ${F_\eta }$ completely distributive? Question 1 was answered recently by Medvedev, and both $1$ and $2$ by Arora and McCleary, using Conrad’s representation of ${F_\eta }$ via right orderings of the free group ${G_\eta }$. Here we answer all four questions by using a completely different tool: The (faithful) representation of ${F_\eta }$ as an $o{\text {-}}2$-transitive $l$-permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite $\eta$, and is here extended to all $\eta > 1$. Curiously, the existence of a transitive representation for ${F_\eta }$ implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of ${G_\eta }$ which suffices all by itself to give a faithful representation of ${F_\eta }$. For finite $\eta$, we find that every transitive representation of ${F_\eta }$ can be made from a pathologically $o{\text {-}}2$-transitive representation by blowing up the points to $o$-blocks; and every pathologically $o{\text {-}}2$-transitive representation of ${F_\eta }$ can be extended to a pathologically $o{\text {-}}2$-transitive representation of ${F_{{\omega _0}}}$.