# Talk: "Recognizing Z-extensions" (J. Delgado, UPC)

We are interested in algorithmically recognizing families of finitely presented groups.

More precisely, if F is one such family, we shall consider the following algorithmic problems:

- Given an arbitrary finite presentation, decide whether it presents a group in F (membership problem for F).

- Given a finite presentation of a group in F, compute ``nice presentations'' for it (enumeration problems within F).

- Given two finite presentations of groups in F, decide whether they are isomorphic or not (isomorphism problem within F).

We will see that the family of finitely presented extensions by Z (i.e. finitely presented groups having Z as quotient) provides both positive and negative answers for some of the stated problems.

In particular, one of the positive answers allows us to reduce the isomorphism problem for unique Z-extensions of a finitely presented group G, to the ``semi-conjugacy'' problem for Out G; while one of the negative answers provides a direct proof for the undecidability of the BNS invariant (a nice geometric invariant long agreed to be hard to compute).