Journal article

VIRTUAL PROPERTIES OF CYCLICALLY PINCHED ONE-RELATOR GROUPS

Gilbert Baumslag, Benjamin Fine, Charles F Miller, Douglas Troeger

INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION | WORLD SCIENTIFIC PUBL CO PTE LTD | Published : 2009

Abstract

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-ext..

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