Kourovka Notebook Problem 21.84

Definitions and mathematical statement

For an integer $d\ge1$, let $G=\operatorname{SL}_d(\mathbb Z)$ be the group of $d$ by $d$ integer matrices of determinant one. The matrix size $d$ is fixed while the permutation degree below tends to infinity. Let $S_n$ act on $[n]=\{1,\ldots,n\}$, with normalized Hamming distance \[ d_n(\sigma,\tau)=n^{-1}|\{i\in[n]:\sigma(i)\ne\tau(i)\}|. \] An almost-homomorphism is a sequence of maps $f_n:G\to S_n$ such that \[ d_n(f_n(g)f_n(h),f_n(gh))\longrightarrow0 \quad\text{for every fixed }g,h\in G. \]

For $m\ge n$, $\sigma\in S_n$, and $\tau\in S_m$, identify $[n]\subseteq[m]$ and put \[ d_n^{\mathrm{flex}}(\sigma,\tau) =n^{-1}\left(|\{i\in[n]:\sigma(i)\ne\tau(i)\}|+m-n\right). \] Is $\operatorname{SL}_d(\mathbb Z)$ flexibly permutation-stable? Explicitly, does every almost-homomorphism admit integers $m_n\ge n$ and genuine homomorphisms $\rho_n:G\to S_{m_n}$ such that \[ d_n^{\mathrm{flex}}(f_n(g),\rho_n(g))\longrightarrow0 \qquad\text{for every fixed }g\in G? \] The problem asks this for the matrix groups as $d$ varies; the case $d=1$ is the trivial boundary case. Flexible convergence requires both approximation on the original points and $(m_n-n)/n\to0$. Thus the correction may enlarge the underlying sets by a vanishing relative amount, while satisfying the group law exactly. No uniformity over all group elements is imposed on these pointwise limits.

Short English statement

Can every approximate finite permutation action of $\operatorname{SL}_d(\mathbb Z)$ be corrected after adding only a vanishing proportion of extra points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2593 (KOU-21.84), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.84. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.