21.84 (2026)

For $\sigma \in S_n$ and $\tau \in S_m$, where $n \leqslant m$, let
$$d_n^{\text{flex}}(\sigma, \tau) = (1/n) \cdot (|\{x \in \{1, \dots, n\} \mid \sigma(x) \neq \tau(x)\}| + (m - n)).$$ An almost-homomorphism $\{f_n\}$ is said to be flexibly close to a homomorphism if there is a sequence of group homomorphisms $\rho_n : G \to S_{m_n}$ with $n \leqslant m_n$ such that $d_n^{\text{flex}}(\rho_n(g), f_n(g)) \to 0$ as $n \to \infty$ for all $g \in G$. The group $G$ is said to be flexibly permutation-stable if every almost-homomorphism of $G$ is flexibly close to a homomorphism.

Is $\text{SL}_n(\mathbb{Z})$ flexibly permutation-stable?