Question 2.9 — Fix a basis for Fn and let An−1⊂ Aut(Fn) be the copy of Aut(Fn−1) corre- sponding to the first n− 1 basis elements.

Definitions and mathematical statement

Fix a free basis $x_1,\ldots,x_n$ of $F_n$. Let $A_{n-1}\leq\operatorname{Aut}(F_n)$ be the embedded $\operatorname{Aut}(\langle x_1,\ldots,x_{n-1}\rangle)$, extending each automorphism by fixing $x_n$. The question is whether, for every group $G$ and homomorphism $\phi:\operatorname{Aut}(F_n)\to G$, \[ |\phi(A_{n-1})|<\infty\quad\Longrightarrow\quad |\phi(\operatorname{Aut}(F_n))|<\infty. \] The conclusion concerns the image, not all of $G$, and no uniform image-size bound is requested.

The rank range is not printed. In rank $2$, $A_1\cong\operatorname{Aut}({\mathbb{Z}})$ is finite, while $\operatorname{Aut}(F_2)$ is infinite, so the identity map shows that a restriction is necessary. The source also reports a positive block-subgroup analogue for $\operatorname{GL}(n,{\mathbb{Z}})$, related to algebraic $K$-theory; its rank hypotheses are likewise omitted.

The continuation discusses a different approach to property (T): Zuk [85] gives a sufficient link-Laplacian spectral criterion for a simply connected group complex. It suggests applying such a criterion to natural $\operatorname{Aut}(F_n)$ or $\operatorname{Out}(F_n)$ complexes, such as the Outer-space spine, but reports David Fisher's observation that Izeki and Natayani [55], alternatively unpublished Schoen--Wang work, obstruct that strategy. The precise complex, Laplacian normalization and obstruction hypotheses are absent. This is a limitation of the proposed method, not a claim that the groups cannot have property (T).

Short English statement

Does finite image on the standard rank-$n-1$ automorphism subgroup force finite image on all of $\operatorname{Aut}(F_n)$, in the intended rank range?

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 2.9 and following spectral-method discussion, supplied PDF page 331. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Zuk [85]; Izeki and Natayani [55] (source spelling); alternative Schoen--Wang work marked unpublished, with the obstruction observation credited to David Fisher. Complete bibliography and exact hypotheses are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000277$, exact source code catalogue entry; supplied OPEN-TRIAGE summary says truncated.