An even better representation for free lattice-ordered groups

Author:
Stephen H. McCleary

Journal:
Trans. Amer. Math. Soc. **290** (1985), 81-100

MSC:
Primary 06F15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0787956-9

MathSciNet review:
787956

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Abstract: The free lattice-ordered group (of rank ) has been studied in two ways: via the Conrad representation on the various right orderings of the free group (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 -transitive -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 on which the action of is both faithful and pathologically -transitive. This 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 of prime subgroups of . All 's with are -isomorphic. This common root system has only four kinds of branches (singleton, three-element, and ), each of which occurs times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0787956-9

Keywords:
Free lattice-ordered group,
ordered permutation group,
right ordered group

Article copyright:
© Copyright 1985
American Mathematical Society