An Even Better Representation for Free Lattice - Ordered Groups
نویسندگان
چکیده
The free lattice-ordered group Fn (of rank r¡) has been studied in two ways: via the Conrad representation on the various right orderings of the free group Gv (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-2-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 (Gn, <) on which the action of Fv is both faithful and pathologically o-2-transitive. This (Cn, <) 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 F. All á^'s with 1 < tj < oo are o-isomorphic. This common root system &j; has only four kinds of branches (singleton, three-element, &j;, and ^„ ), each of which occurs 2"° times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of &j.; 1. The Conrad representation has been used most extensively by Arora and McCleary [1], who studied centralizers of certain elements of F. The GlassMcCleary representation was exploited (and partially developed) by McCleary in [7], to which the present paper is a sequel. Familiarity with [7] (but not [1]) is assumed. The Conrad representation [2] proceeds as follows: Given any right ordering (Gv, <) of the free group Gv, the right regular representation vp^ i j ' ' j We shall refer to this as the natural action of Fn on (G^, <). When this action is a representation (i.e., faithful), we shall call (Gv, <) a representing right ordering. Another kind of action of Fv is usual transitive action on the chain F /P of right cosets of a prime subgroup P (namely (Pf)w = P(fw)). When this action is a representation, P is called a representing subgroup of F . Received by the editors February 21, 1984. 1980 Mathematics Subject Classification. Primary 06F15.
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