Removing the K-regularity assumption

Determine whether the K-regularity assumption in Theorem 1 can be removed from the centralizer-rigidity theorem for C^1-small, volume-preserving perturbations of non-trivial diagonal maps on compact quotients of SL_n(R).

Background

The main theorem assumes that the perturbed centralizer action is K-regular, providing uniform quantitative control of the action near the identity. The authors explain that this hypothesis is used in constructing an adapted Lie-algebra isomorphism between the perturbed and algebraic centralizers.

The unresolved issue is whether such quantitative control follows intrinsically from the structure of the centralizer, possibly after passing to a finite-index subgroup or choosing suitable coordinates on center leaves, thereby making the explicit K-regularity hypothesis unnecessary.

References

Can the $K$-regularity assumption in Theorem \ref{cenrignongen} be removed?

— Local centralizer rigidity for a non-generic diagonal map  (2609.04643 - Wang et al., 4 Sep 2026) in Section 1, subsection “Further questions,” item 1

What happens for the exceptional low-rank one-root cases $\operatorname{diag}(et,et,e{-2t})\in \mathrm{SL}_3(\mathbb R)$ and $\operatorname{diag}(et,et,e{-t},e{-t}) \in \mathrm{SL}_4(\mathbb R)?$

— Local centralizer rigidity for a non-generic diagonal map  (2609.04643 - Wang et al., 4 Sep 2026) in Section 1, subsection “Further questions,” item 3

For example, can one treat the case of $f_0=L_a: \mathrm{SL}{2n}R/\Gamma\to \mathrm{SL}{2n}R/\Gamma$ , where $a=\begin{pmatrix} eR_1&0\0& e{-1}R_2 \end{pmatrix}$ and $R_1,R_2\in \mathrm{SO}(n)$?

— Local centralizer rigidity for a non-generic diagonal map  (2609.04643 - Wang et al., 4 Sep 2026) in Section 1, subsection “Further questions,” item 4