Algorithmic recognition of infinite cyclic extensions

TL;DR

This paper proves the undecidability of certain algorithmic problems related to infinite cyclic extensions and their invariants, highlighting fundamental limits in computational group theory.

Contribution

It establishes the undecidability of recognizing finitely generated bases in $bZ$-extensions and links the isomorphism problem to the semi-conjugacy problem for outer automorphisms.

Findings

Undecidability of recognizing finitely generated bases in $bZ$-extensions

Undecidability of the BNS invariant

Equivalence between isomorphism and semi-conjugacy problems

Abstract

We prove that one cannot algorithmically decide whether a finitely presented $\mathbb{Z}$-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique $\mathbb{Z}$-extensions, and the semi-conjugacy problem for deranged outer automorphisms.