Kourovka Notebook Problem 21.61

Definitions and mathematical statement

Fix an integer $n\ge1$, the free group $F_n$ on a specified basis $x_1,\ldots,x_n$, and an automorphism $\varphi$ of $F_n$. The associated free-by-cyclic group is \[ G=F_n\rtimes_\varphi\mathbb Z =\langle x_1,\ldots,x_n,t\mid tx_it^{-1}=\varphi(x_i)\ (1\le i\le n)\rangle. \] The words $\varphi(x_i)$ and this generating alphabet are fixed as part of the ambient group, rather than supplied afresh as variable inputs. Elements of $G$ are specified by finite words in these generators and their inverses.

Is there, for every such fixed $G$, an algorithm that takes any finite list of words $s_1,\ldots,s_k$ and outputs a finite presentation for $H=\langle s_1,\ldots,s_k\rangle\le G$? A finite presentation consists of finitely many formal generators and finitely many relator words; the quotient of their free group by the normal closure of those relators must be isomorphic to $H$.

The algorithm must halt on every input list, including lists with redundant generators or relations among their entries. It may use a presentation on new generators: the question asks for a finite presentation of the subgroup, not a prescribed choice of generators. The number and lengths of the input words are unrestricted. One algorithm must work for every finite subset inside the fixed ambient group. The assertion does not impose a complexity bound or require that a single algorithm work uniformly from arbitrary input automorphisms $\varphi$.

Short English statement

For every fixed finitely generated free-by-cyclic group, can one algorithm compute a finite presentation for each subgroup given by finitely many words?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2570 (KOU-21.61), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.61. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.