Question 2.6 — If there is a homomorphism from a subgroup of finite index in Aut(Fn) onto a subgroup of finite index in GL(m, Z), th…

Definitions and mathematical statement

Let $F_n$ be the free group of rank $n$, and let $\operatorname{GL}(m,{\mathbb{Z}})$ be the group of invertible integral $m\times m$ matrices. The source asks whether \[ \begin{gathered} H\leq\operatorname{Aut}(F_n),\quad [\operatorname{Aut}(F_n):H]<\infty,\\ K\leq\operatorname{GL}(m,{\mathbb{Z}}),\quad [\operatorname{GL}(m,{\mathbb{Z}}):K]<\infty,\quad \exists\text{ surjection }H\twoheadrightarrow K \end{gathered} \quad\Longrightarrow\quad m\geq n-1. \] No injectivity of the homomorphism is requested. The precise ranges of $n,m$ are not supplied. In particular $m=1$ gives the finite group $\operatorname{GL}(1,{\mathbb{Z}})=\{\pm1\}$, with the trivial subgroup already of finite index; this makes an exclusion or infinite-target convention necessary when $n\geq3$. Such a restriction is not silently added. The closing phrase "Indeed one might ask" refers to an omitted further question.

Short English statement

Must a finite-index quotient of a finite-index subgroup of $\operatorname{Aut}(F_n)$ inside $\operatorname{GL}(m,{\mathbb{Z}})$ satisfy $m\geq n-1$, with the missing low-rank exclusions clarified?

Sources

• M. Bridson and K. Vogtmann, chapter in Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 2.6, supplied PDF page 330. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. No additional reference is supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000274$, exact source code catalogue entry; supplied OPEN-TRIAGE summary says truncated.