Questions in Geometric Group Theory — Q 12.6

Definitions and mathematical statement

Fix a free basis $A=\{a_1,\ldots,a_n\}$ of the rank-$n$ free group $F_n$, with $n\geq2$, and let $|w|_A$ be the length of the freely reduced word representing $w$. An automorphism $\alpha$ is supplied by the reduced words $\alpha(a_i)$. Use the explicit input-size measure \[ L(\alpha)=\max_{1\leq i\leq n}|\alpha(a_i)|_A. \] Its fixed subgroup is $\operatorname{Fix}(\alpha)=\{g\in F_n:\alpha(g)=g\}$. Assume it is nontrivial cyclic, say $\operatorname{Fix}(\alpha)=\langle u\rangle$, where $u$ is one of the two generators of that infinite cyclic group.

Find an explicit effective upper bound $B(n,L)$ such that \[ |u|_A\leq B(n,L(\alpha)) \] for every such automorphism. The length is based reduced length, not the shortest length among conjugates of $u$. Conjugating a subgroup can change the former length substantially.

The source records algorithmic computability of fixed subgroups, so the issue here is quantitative control in terms of input complexity, rather than merely an existence proof for some computable bound obtained by exhaustive enumeration. One may separately seek bounds in terms of a relative train-track representative, as suggested in the source. Such a bound should specify its complexity parameters and how they are controlled by the supplied automorphism words.

Short English statement

How long can a generator of a cyclic fixed subgroup be, in terms of the lengths describing the free-group automorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101206 (AMR-010-1206), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 12.6 (PDF page 20). Original source indexed by AMR-010-1206. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.