Questions in Geometric Group Theory — Q 3.9

Definitions and mathematical statement

For an integer $n\geq2$, let $F_n=\langle a_1,\ldots,a_n\rangle$ be the free group of rank $n$. An ordered tuple $(x_1,\ldots,x_n)$ is a basis when the homomorphism $F_n\to F_n$ sending $a_i$ to $x_i$ is an automorphism. Put \[ B_n=\{(x_1,\ldots,x_n)\in F_n^n:(x_1,\ldots,x_n)\text{ is a basis of }F_n\}. \] Use the first-order language of groups $(\cdot,{}^{-1},1)$. The set $B_n$ is definable without parameters if there is a formula $\varphi(x_1,\ldots,x_n)$ in this language whose realizations in $F_n$ are exactly $B_n$. It is definable with parameters if there are a finite tuple $c\in F_n^m$ and a formula $\varphi(x_1,\ldots,x_n;y_1,\ldots,y_m)$ with \[ B_n=\{x\in F_n^n:F_n\models\varphi(x;c)\}. \] Source conjecture. For each integer $n>1$, $B_n$ is not definable. The displayed source sentence does not say whether parameters are permitted, so its two interpretations must be distinguished.

The supplied research summary reports: without parameters, $B_n$ is not definable for every $n\geq2$; with parameters, $B_2$ is definable whereas $B_n$ is not definable for every $n\geq3$. For rank two it attributes parameter-definability to Nielsen's criterion that $[g,h]$ be conjugate to $[a_1,a_2]$ or its inverse, where $[u,v]=u^{-1}v^{-1}uv$. Thus the two interpretations have different answers at rank two. These are reported literature conclusions, not independently verified here.

Short English statement

Can first-order group formulas describe precisely the ordered free bases of $F_n$? The supplied resolution distinguishes formulas with parameters from formulas without them, with a rank-two exception only in the former case.

Sources

• M. Bestvina, Questions in Geometric Group Theory (2004), Question 3.9, PDF page 11: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf; manual transcription, accessed 2026-07-29.
• Supplied research-summary attributions: Nielsen (1917), rank-two commutator criterion; Pillay, characterization of the generic type; Sklinos, bigness of the non-primitives; Bestvina--Feighn, negligibility or co-negligibility of definable sets and the higher-rank primitive-set obstruction. Apart from Nielsen's year, no publication identifiers are supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $1100309$, source code catalogue entry, Questions in Geometric Group Theory --- Q 3.9. The record labels the question solved and SOLVED-IN-LITERATURE; the source-list background separately retains NEEDS_REVIEW status and rights qualifications.