Question 3.6 — Is there a map Out(Fn)→ Out(Fm) that induces an isomorphism on homology in the stable range?

Definitions and mathematical statement

Let $F_n$ be the rank-$n$ free group and $\operatorname{Out}(F_n)$ its outer automorphism group. The preceding local source item 11000285 recalls integral homology stability: for each degree $i$, $H_i(\operatorname{Out}(F_n);{\mathbb{Z}})$ is independent of $n$ once the rank is sufficiently large. The excerpt does not supply a stability bound.

Question 3.6. Is there a group homomorphism \[ \phi:\operatorname{Out}(F_n)\longrightarrow\operatorname{Out}(F_m) \] that induces isomorphisms on integral homology in the common stable range? In a specified range of degrees, the target is that the same $\phi$ induce \[ \phi_*:H_i(\operatorname{Out}(F_n);{\mathbb{Z}})\xrightarrow{\cong} H_i(\operatorname{Out}(F_m);{\mathbb{Z}}) \] for all degrees in that range. The surrounding discussion concerns passing to larger rank, but the excerpt does not say for which $m>n$, or whether some sufficiently large $m$ suffices. Allowing $m=n$ and the identity would trivialize the intended question. No injectivity of $\phi$ is requested, and abstract equality of stable homology groups is not the construction of such a group map. The rational-stability report in the summary is weaker than the integral context supplied by the source.

The appended paragraph starts a separate quotient discussion. Here the letter $m$ is reused as a positive power parameter, rather than the target rank above. Let $\lambda\in\operatorname{Aut}(F_n)$ send $a_1$ to $a_2a_1$ and fix the other basis elements, and define \[ Q(n,m)=\operatorname{Aut}(F_n)/\langle\!\langle\lambda^m\rangle\!\rangle. \] The source notes that Nielsen moves of this type are conjugate, making the chosen basis immaterial. It contrasts the resulting quotients with matrix congruence quotients, identifying the quotient by powers of elementary matrices with $\mathrm{GL}_n({\mathbb{Z}}/m)$. That exact matrix-quotient assertion needs rank and reduction-image qualifications; reduction of $\mathrm{GL}_n({\mathbb{Z}})$ need not give all of $\mathrm{GL}_n({\mathbb{Z}}/m)$ without further conditions.

It reports that Bridson--Vogtmann [21] prove $Q(n,m)$ infinite for sufficiently large $m$ by obtaining a quotient that contains the free Burnside group \[ B(n-1,m)=F_{n-1}/\langle\!\langle w^m:w\in F_{n-1}\rangle\!\rangle. \] The exact rank and exponent hypotheses are not reproduced. The claim is about a subgroup of a quotient of $Q(n,m)$, not necessarily a subgroup of $Q(n,m)$ itself. The source also refers to alternatives from Ol'shanskii [73], subsection 39.3, and introduces a following question absent from this record. This quotient material is not a solution to the stable-homology-map problem.

Short English statement

Construct a map between different-rank free-group outer automorphism groups that realizes stable integral homology isomorphisms. The source separately introduces quotients obtained by killing powers of Nielsen moves; their properties do not themselves provide the desired map.

Sources

• Local UnsolvedMath registry, permanent ID 11000286, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, chapter "Automorphism groups of free, surface and free abelian groups," Question 3.6, PDF page 333; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• Adjacent local ID 11000285 supplies the integral-stability context and source labels [50,51]. The present text cites Bridson--Vogtmann [21] and A. Yu. Ol'shanskii [73], subsection 39.3. Full bibliography entries and theorem hypotheses are absent.