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On Almost Primitive Elements of Free Groups With an Application to Fuchsian Groups

Published online by Cambridge University Press:  20 November 2018

A. M. Brunner ,
R. G. Burns  and
Sheila Oates-Williams
A. M. Brunner
Affiliation:
Department of Mathematics, University of Wisconsin-Parkside, Kenosha, Wisconsin, U.S.A.53141
R. G. Burns
Affiliation:
Department of Mathematics, York University, North York, Ontario, M3J 1P3
Sheila Oates-Williams
Affiliation:
Department of Mathematics, The University of Queensland,Queensland, Australia 4072
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Abstract

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An element of a free group F is called almost primitive in F, if it is primitive in every proper subgroup containing it, though not in F itself. Several examples of almost primitive elements (APEs) are exhibited. The main results concern the behaviour of proper powers wℓ of certain APEs w in a free group F (and, more generally, in free products of cycles) with respect to any subgroup H containing such a power “minimally“: these assert, in essence, that either such powers of w behave in H as do powers of primitives of F, or, if not, then they “almost” do so and furthermore H must then have finite index in F precisely determined by the smallest positive powers of conjugates of w lying in H. Finally, these results are applied to show that the groups of a certain class (potentially larger than that of finitely generated Fuchsian groups) have the property that all their subgroups of infinité index are free products of cyclic groups.

Keywords

20E0520F38
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 2 , 01 April 1993 , pp. 225 - 254
Copyright
Copyright © Canadian Mathematical Society 1993

References

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