Abstract
Funding: Acknowledgements. The authors would also like to thank the referee for their very careful and thorough reading of the paper. This publication is in part a product of a visit of the first and third author to the Mathematisches Forschungsinstitut Oberwolfach, Germany in December 2016 as part of their Research In Pairs program. The third author was partially supported by NSF grants DMS–1600635 and DMS-1854367.
Highlights
Subgroups of P L+(I), the group of order preserving, piecewise linear self homeomorphisms of the unit interval, have been a source of groups with interesting properties in which calculations are practical
The elementary amenability class (EA-class) of a group G in EG is an ordinal valued measure of the complexity of the recursive construction of G. (Details are given in Section 3.) Thompson’s group F is not elementary amenable; it is finitely generated and every nontrivial normal subgroup of F contains isomorphic copies of F
While we do not settle Conjecture 1, this paper explores the universe of F -less subgroups of P L+(I) and finds a complex collection S of elementary amenable subgroups of Thompson’s group F itself
Summary
Subgroups of P L+(I), the group of order preserving, piecewise linear self homeomorphisms of the unit interval, have been a source of groups with interesting properties in which calculations are practical. (Details are given in Section 3.) Thompson’s group F is not elementary amenable; it is finitely generated and every nontrivial normal subgroup of F contains isomorphic copies of F (see [8]). It contains a review of a number of prerequisites for the paper: details from [3]; ordinals and their arithmetic; elementary amenable groups and EA-class; wreath products of permutation groups. This analysis is used to show that (S, →) is a well-order with ordertype ε0, where the proof of Theorem 1 is completed.
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