An Even Better Representation for Free Lattice-Ordered Groups

Abstract

The free lattice-ordered group ${F_\eta }$ (of rank $\eta$) has been studied in two ways: via the Conrad representation on the various right orderings of the free group ${G_\eta }$ (sharpened by Kopytov’s observation that some one right ordering must by itself give a faithful representation), and via the Glass-McCleary representation as a pathologically $o{\text {-}}2$-transitive $l$-permutation group. Each kind of representation yields some results which cannot be obtained from the other. Here we construct a representation giving the best of both worlds—a right ordering $({G_\eta }, \leqslant )$ on which the action of ${F_\eta }$ is both faithful and pathologically $o{\text {-}}2$-transitive. This $({G_\eta }, \leqslant )$ has no proper convex subgroups. The construction is explicit enough that variations of it can be utilized to get a great deal of information about the root system ${\mathcal {P}_\eta }$ of prime subgroups of ${F_\eta }$. All ${\mathcal {P}_\eta }$’s with $1 < \eta < \infty$ are $o$-isomorphic. This common root system ${\mathcal {P}_f}$ has only four kinds of branches (singleton, three-element, ${\mathcal {P}_f}$ and ${\mathcal {P}_{{\omega _0}}}$), each of which occurs ${2^{{\omega _0}}}$ times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of ${\mathcal {P}_f}$.