Existence of a Carnot group with noninteger curvature exponent

Establish the existence of a Carnot group with a noninteger curvature exponent, as conjectured in Conjecture 18 of Zhang (2025).

Background

The paper proves that integer-parameter b1-Grushin planes provide real-analytic sub-Riemannian structures with finite, noninteger curvature exponents. It then places this result in the broader context of Carnot groups, where the existence of a group with a noninteger curvature exponent has been conjectured but remains unresolved.

The authors note that the classical Grushin plane is a quotient of the Heisenberg group and that both have curvature exponent 5. For higher integer b1, the corresponding b1-Grushin plane is a quotient of a model filiform Carnot group, but that filiform group fails every measure contraction property. Thus, this natural quotient construction cannot settle the conjecture.