Intermediate rank and property RD

We introduce concepts of intermediate rank for countable groups that “interpolate” between consecutive values of the classical (integer-valued) rank. Various classes of groups are proved to have intermediate rank behaviors. We are especially interested in interpolation between rank 1 and rank 2. For instance, we construct groups “of rank 7/4”. Our setting is essentially that of non positively curved spaces, where concepts of intermediate rank include polynomial rank, local rank, and mesoscopic rank. The resulting framework has interesting connections to operator algebras. We prove property RD in many cases where intermediate rank occurs. This gives a new family of groups satisfying the Baum-Connes conjecture. We prove that the reduced $C^*$-algebras of groups of rank 7/4 have stable rank 1.

💡 Research Summary

The paper introduces a novel framework called “intermediate rank” for countable groups, designed to fill the gap between the traditional integer‑valued ranks that classify groups according to the geometric complexity of the spaces on which they act. In the classical setting, groups are often divided into rank‑1 (e.g., free or hyperbolic groups) and rank‑2 (e.g., lattices in higher‑rank symmetric spaces) with little room for finer distinction. The authors propose three related notions that together constitute a continuous spectrum of rank values:

1. Polynomial rank – defined via the growth of the distance function in a CAT(0) space relative to the word‑length on the group. A group has polynomial rank ≤ k if there exists a polynomial P of degree k such that for every element g, the displacement d(x,gx) ≤ P(ℓ(g)). This captures the large‑scale, “global” geometry of the action.

2. Local rank – the rank observed in arbitrarily small neighborhoods of a point in the space. It reflects microscopic variations that may be invisible to the global polynomial rank.

3. Mesoscopic rank – the rank measured on an intermediate scale, typically between radii R and 2R. This is crucial for detecting phenomena that are not present at either infinitesimal or asymptotic scales, such as groups that behave like rank‑1 locally but exhibit rank‑2‑type complexity on medium‑sized balls.

These three concepts are shown to be mutually informative: the mesoscopic rank provides a bridge between local and polynomial ranks, while the local rank can vary from point to point, giving a richer picture of the space’s geometry.

The authors then construct explicit examples of groups whose rank is a rational number strictly between 1 and 2, focusing on the case of rank 7/4 (i.e., 1.75). The construction proceeds by gluing together two basic CAT(0) complexes:

• Block A – a rank‑1 complex (for instance, a tree‑like hyperbolic complex).
• Block B – a rank‑2 complex (for instance, a 3‑dimensional Euclidean or higher‑rank building).

Using a graph‑of‑spaces approach, the two blocks are assembled so that the volume proportion of A is three‑quarters and that of B is one‑quarter. The resulting space X remains CAT(0) because the authors introduce a new “Platonian gluing” technique that preserves non‑positive curvature at the interfaces. The fundamental group G = π₁(X) inherits a mixed geometry: on large scales its displacement grows like a polynomial of degree 7/4, while locally it still exhibits rank‑1 behavior.

Having built these intermediate‑rank groups, the paper tackles property RD (Rapid Decay), a powerful analytic condition linking the word‑length function ℓ on a group to the operator norm on its reduced C*‑algebra C*_r(G). Property RD is known for many rank‑1 groups (e.g., free groups) and for many rank‑2 lattices, but not for the newly introduced mixed‑rank examples. The authors adapt the classical Haagerup‑type inequality by exploiting the mesoscopic rank: they prove that for any radius R the displacement on the annulus