Some Questions — Question 47

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $F_n$ be a free group of finite rank $n\geq2$. A pseudocharacter is a homogeneous quasimorphism $q:F_n\to\mathbb R$: its defect \[D(q)=\sup_{g,h\in F_n}|q(gh)-q(g)-q(h)| \] is finite and $q(g^k)=kq(g)$ for all $g\in F_n$ and integers $k$. It is automorphism invariant when $q(\alpha(g))=q(g)$ for every $\alpha\in\operatorname{Aut}(F_n)$ and $g\in F_n$.

Statement recorded in the supplied source.

Does $F_n$ admit a nonzero automorphism-invariant pseudocharacter for each $n\geq2$? The cited 2023 theorem answers this historical question affirmatively and in fact gives an infinite-dimensional space of such functions. [S2]

Short English statement

Can a nonzero bounded-defect additive function on a finite-rank free group be invariant under every automorphism? The cited theorem gives an affirmative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1200047 (AMR-011-0047). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 F. Fournier-Facio and R. D. Wade, Aut-invariant quasimorphisms on groups, arXiv:2211.00800v2 (2023), §1, Question A (Abért, 2010, Question 47) and Theorem B. https://arxiv.org/abs/2211.00800.