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Pathological and highly transitive representations of free groups. (English) Zbl 1342.06012

As we know, S. H. McCleary has shown how to represent any free lattice ordered group (free \(l\)-group) as a permutation group of an ordered set [Trans. Am. Math. Soc. 290, 69-79 (1985; Zbl 0546.06013)]. P. J. Cameron constructed a copy of the free group of rank \(2^{\aleph_0}\) within \(\operatorname{Aut}(\mathbb Q)\), moreover, he has shown that any doubly transitive automorphism group of a linear order must contain a copy of \(\operatorname{Aut}(\mathbb Q)\) [Permutation groups. Cambridge: Cambridge University Press (1999; Zbl 0922.20003)].
In this paper, the author shows that the full versatility of doubly transitive automorphism groups is not necessary by extending Cameron’s construction to a larger class of permutation groups and the author generalizes his result by constructing pathological (permutations of unbounded support) and \(\omega\)-transitive (highly transitive) representations of free groups. In particular, and working solely within ZFC, the author shows that any large subgroup of \(\operatorname{Aut}(\mathbb Q)\) (resp. \(\operatorname{Aut}(\mathbb R)\)) contains an \(\omega\)-transitive and pathological representation of any free group of rank \(\lambda\in[\aleph_0,2^{\aleph_0}]\) (resp. of rank \(2^{\aleph_0}\)). Assuming the continuum to be a regular cardinal, the author shows that pathological and \(\omega\)-transitive representations of uncountable free groups abound within large permutation groups of linear orders. Lastly, the author also finds a bound on the rank of free subgroups of certain restricted direct products.

MSC:

06F15 Ordered groups
20E05 Free nonabelian groups
03E75 Applications of set theory
06A05 Total orders
20B27 Infinite automorphism groups

Software:

PlanetMath
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References:

[1] Bennett, C.D.: Explicit free subgroups of Aut(R, ≤). Proc. Amer. Math. Soc. 125(5), 1305-1308 (1997). MR 1363412 (97g:06020) · Zbl 0865.06009
[2] Bludov, V.V., Droste, M., Glass, A.M.W.: Automorphism groups of totally ordered sets: a retrospective survey. Math. Slovaca 61(3), 373-388 (2011). MR 2796250 · Zbl 1265.06048
[3] Bludov, V.V., Glass, A.M.W.: Right orders and amalgamation for lattice-ordered groups. Math. Slovaca 61(3), 355-372 (2011). MR 2796249 · Zbl 1265.06049
[4] Cameron, P.J.: Oligomorphic permutation groups. Lond. Math. Soc. Stud. Texts 45 (1999) · Zbl 0813.20002
[5] Cohn, P.M.: Groups of order automorphisms of ordered sets. Mathematika 4, 41-50 (1957). MR 0091280 (19,940e) · Zbl 0088.02702
[6] Conrad, P.: Free lattice-ordered groups. J. Algebra 16, 191-203 (1970). MR 0270992 (42 #5875) · Zbl 0213.31502
[7] de la Harpe, P.: Topics in geometric group theory. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2000). MR 1786869 (2001i:20081) · Zbl 0965.20025
[8] Droste, M., Holland, W.C., Macpherson, H.D.: Automorphism groups of infinite semilinear orders. I II. Proc. Lond. Math. Soc. (3) 58(3), 454-478, 479-494 (1989). MR 988099 (90b:20006) · Zbl 0636.20003
[9] Glass, A.M.W.: l-simple lattice-ordered groups. Proc. Edinb. Math. Soc. (2) 19(2), 133-138 (1974/75). MR 0409309 (53 #13069) · Zbl 0296.06010
[10] Holland, C.: The lattice-ordered groups of automorphisms of an ordered set. Michigan Math. J. 10, 399-408 (1963). MR 0158009 (28 #1237) · Zbl 0116.02102
[11] Kunen, K.: Set theory. Studies in Logic and the Foundations of Mathematics, vol. 102. Amsterdam, North-Holland Publishing Co. (1983), An introduction to independence proofs, Reprint of the 1980 original MR 756630 (85e:03003) · Zbl 0534.03026
[12] McCleary, S.H.: Free lattice-ordered groups represented as o- 2 transitive l-permutation groups. Trans. Amer. Math. Soc. 290(1), 69-79 (1985). MR 787955 (86m:06034a) · Zbl 0546.06013
[13] Weiss, U.: Ping-Pong Lemma (version 5), PlanetMath.org
[14] White, S.: The group generated by x ↦ x + 1 and x ↦ xp is free. J. Algebra 118(2), 408-422 (1988). MR 969681 (90a:12014) · Zbl 0662.20024
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