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Talk: Solvability and unsolvability results for certain cyclic extensions (J. Delgado, UPC)

X Congreso en Teoría de Grupos

IMUS (Sevilla)

Abstract:

We are interested in the algorithmic recognition of certain classes of groups. Concretely, we will study (finitely presented) extensions by Z, i.e. finitely presented groups that have a normal subgroup G with quotient isomorphic to Z. Such extensions are, of course, semidirect products of the form GαZ, where α is an automorphism of the base group G. It turns out that finitely presented extensions by Z are not necessarily extensions by Z of finitely presented groups ([f.p.]-by-Z groups, for short); but when they are (i.e. when G is finitely presented) we can compute defining isomorphisms α, and we can give necessary and sufficient conditions on α for such a GαZ extension to be unique. This allows us to reduce the isomorphism problem for unique extensions by Z of a finitely presented group G to the "semi-conjugacy" problem for OutG. On the other hand, we prove that the problem of deciding whether an arbitrary finite presentation is [f.p.]-by-Z (or [f.g.]-by-Z) is undecidable.