Finite asymptotic dimension of bounded-to-one commutative monoid actions

Determine whether every finitely generated commutative monoid acting bounded-to-one on a set has finite asymptotic dimension, and whether its asymptotic dimension is at most the torsion-free rank of the monoid's group completion.

Background

The paper proves finite Borel asymptotic dimension for free bounded-to-one actions of finitely generated commutative monoids. For non-free actions, the authors establish equality between classical and Borel asymptotic dimension, but do not establish finiteness of the classical quantity. The unresolved issue is therefore whether stabilizers and other non-free behavior can force infinite asymptotic dimension, and, more sharply, whether the natural rank bound remains valid beyond free actions.

References

That is, the following is open (this is restated later as \zcref{qn:classical_asdim}): Let M be a finitely generated commutative monoid and let M\curvearrowright X be a (bounded-to-one) M-set. Must we have \asdim(M\curvearrowright X) < \infty?

— Hyperfiniteness of bounded-to-one actions of commutative monoids  (2608.18439 - Shinko et al., 19 Aug 2026) in Question 1.4, Section 1; restated as Question in Section 3

We were not able to remove the freeness assumption in \zcref{thm:free-fg-asdim-intro}, i.e. to show that all (not necessarily free) actions have finite Borel asymptotic dimension.

— Hyperfiniteness of bounded-to-one actions of commutative monoids  (2608.18439 - Shinko et al., 19 Aug 2026) in Section 1, discussion following the proof of the finitely generated case