Author:
Brunner A. M.,Burns R. G.,Oates-Williams Sheila
Abstract
AbstractAn 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.
Publisher
Canadian Mathematical Society
Cited by
11 articles.
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