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Highly transitive representations of free groups and free products

Published online by Cambridge University Press:  17 April 2009

A.M.W. Glass  and
Stephen H. McCleary
A.M.W. Glass
Affiliation:
Mathematics and Statistics Department, Bowling Green State University, Bowling Green, Ohio 43403, United States of America
Stephen H. McCleary
Affiliation:
Mathematics and Statistics Department, Bowling Green State University, Bowling Green, Ohio 43403, United States of America
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Abstract

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A permutation group is highly transitive if it is n–transitive for every positive integer n. A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every α1 < … < αn and β1 < … < βn in Q there exists g ∈ G such that αig = βi, i = 1, …, n. The free group Fn(2 ≤ η ≤ אo) can be faithfully represented as a highly order-transitive group of order-preserving permutations of Q, and also (reproving a theorem of McDonough) as a highly transitive group on the natural numbers N. If G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G * H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then G * H can be faithfully represented as a highly transitive group on N. Some of the representations of Fη on Q can be extended to faithful representations of the free lattice-ordered group Lη.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 19 - 36
Copyright
Copyright © Australian Mathematical Society 1991

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