Curvature exponent of a Carnot group projecting onto the b1-Grushin plane

Determine whether a Carnot group admitting the b1-Grushin plane as a quotient has curvature exponent equal to N_b1, thereby providing a Carnot-group example with a finite noninteger curvature exponent for integer b1e=2.

Background

The b1-Grushin plane has curvature exponent N_b1, which is irrational for every integer b1e=2. This motivates investigating whether the noninteger exponent can be lifted from the quotient to an appropriate Carnot group.

The paper identifies the natural candidate for integer b1 as the model filiform Carnot group of step b1+1. However, the authors state that these filiform groups fail every measure contraction property when b1e=2, so the obvious quotient construction cannot answer the question. Whether some other Carnot group with the required quotient structure has curvature exponent N_b1 is left unresolved.

References

Since the classical Grushin plane is a quotient of the Heisenberg group and both spaces have curvature exponent 5, it is natural to ask whether the Carnot group admitting \mathbb G_\alpha2 as a quotient might likewise have curvature exponent N_\alpha and provide the example predicted by the conjecture.

— The measure contraction property on Grushin spaces  (2609.10208 - Albert et al., 9 Sep 2026) in Section 1, Introduction