Questions in Geometric Group Theory — Q 12.2

Definitions and mathematical statement

Let $F_n$ be the free group of finite rank $n\geq2$, and let $\operatorname{Out}(F_n)=\operatorname{Aut}(F_n)/\operatorname{Inn}(F_n)$, where inner automorphisms are conjugations by elements of $F_n$. A subgroup $K\leq F_n$ is characteristic if every automorphism of $F_n$ preserves it.

Whenever $K$ is characteristic and has finite index, each automorphism descends to the finite quotient $F_n/K$. Inner automorphisms descend to inner automorphisms, giving a well-defined homomorphism \[ \rho_K:\operatorname{Out}(F_n)\longrightarrow \operatorname{Out}(F_n/K). \] Its kernel is a principal congruence subgroup in the sense of this problem. The congruence subgroup property asks whether every finite-index subgroup $U\leq\operatorname{Out}(F_n)$ contains $\ker\rho_K$ for some such $K$.

Does this hold for every finite rank $n$? Equivalently, do these kernels form a neighborhood basis for the identity in the full profinite topology, whose basic neighborhoods are finite-index normal subgroups? The subgroup $U$ need not be normal, since it contains a finite-index normal core.

The quotient $F_n/K$ may be any finite characteristic quotient; it is not restricted to abelian quotients or reductions of the homology representation to matrices modulo an integer. Replacing all finite characteristic quotients by only those abelian ones would ask a stronger and different question. The chosen quotient may depend on $U$, with no uniform bound on its size required.

Short English statement

Does every finite-index subgroup of $\operatorname{Out}(F_n)$ contain the kernel of its action on some finite characteristic quotient of $F_n$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101202 (AMR-010-1202), 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 12.2 (PDF page 19). Original source indexed by AMR-010-1202. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.