Geodesic
On Almost Primitive Elements of Free Groups With an Application to Fuchsian Groups
Canadian journal of mathematics, Tome 45 (1993) no. 2, pp. 225-254

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1993-011-9
Mots-clés : 20E05, 20F38 
Brunner, A. M.; Burns, R. G.; Oates-Williams, Sheila. On Almost Primitive Elements of Free Groups With an Application to Fuchsian Groups. Canadian journal of mathematics, Tome 45 (1993) no. 2, pp. 225-254. doi: 10.4153/CJM-1993-011-9
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