Groups, Geometry, and Dynamics


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Volume 3, Issue 1, 2009, pp. 1–37
DOI: 10.4171/GGD/50

Published online: 2009-03-31

A minimal non-solvable group of homeomorphisms

Collin Bleak[1]

(1) University of St Andrews, United Kingdom

Let PLo(I) represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group W and show that W embeds in every non-solvable subgroup of PLo(I). We find mild conditions under which other non-solvable subgroups B, (≀ℤ≀), (ℤ≀), and (≀ℤ)) embed in subgroups of Let PLo(I) represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group W and show that W embeds in every non-solvable subgroup of PLo(I). We show that all solvable subgroups of PLo(I) embed in all non-solvable subgroups of PLo(I). These results continue to apply if we replace PLo(I) by any generalized Thompson group Fn.

Keywords: PL homeomorphisms, group actions, unit interval, non-solvable groups, Thompson's group F

Bleak Collin: A minimal non-solvable group of homeomorphisms. Groups Geom. Dyn. 3 (2009), 1-37. doi: 10.4171/GGD/50