Nilpotent groups

Definitions and mathematical statement

For a metric space $(X,d)$ with a Markov kernel $x\mapsto m_x$ whose probability measures have finite first moments, define \[ W_{1,d}(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times X}d(u,v)\,d\pi(u,v),\qquad \kappa(x,y)=1-\frac{W_{1,d}(m_x,m_y)}{d(x,y)}\quad(x\ne y), \] where $\Pi(\mu,\nu)$ is the set of couplings with marginals $\mu,\nu$. This is the coarse Ricci curvature associated with the metric and kernel.

Determine coarse Ricci curvature for discrete or continuous nilpotent groups equipped with the intended metric and random walk. The particular example in the source is the discrete Heisenberg group \[ H=\langle a,b,c\mid ac=ca,\ bc=cb,\ [a,b]=c\rangle. \] Here one may fix $[a,b]=aba^{-1}b^{-1}$ as the commutator convention. A usual meaning of the natural symmetric walk generated by $a,b$ is the word metric for $S=\{a,a^{-1},b,b^{-1}\}$ and the kernel \[ m_g=\frac14\sum_{s\in S}\delta_{gs}. \] Compute the corresponding curvature profile and determine whether its reported negative small-scale curvature tends to zero at larger scales.

A complete limiting proposition requires a specified scale family. For example, if $(d_r,m^{(r)})$ is the intended family, its curvature is \[ \kappa_r(g,h)=1-\frac{W_{1,d_r}(m^{(r)}_g,m^{(r)}_h)}{d_r(g,h)}. \] The source does not state whether scale means changing point separation, taking longer-time transition laws, changing the step radius, or a metric-measure rescaling, nor which pairs or uniformity are involved in the limit. Accordingly it does not identify a unique assertion of the form $\kappa_r\to0$. The metric and kernel for the general discrete and continuous groups are likewise unspecified.

Short English statement

Compute transport-based coarse Ricci curvature for nilpotent groups, especially the natural walk on the discrete Heisenberg group. Make precise and decide whether negative curvature at small scales disappears at large scales.

Sources

• Yann Ollivier, Discrete Ricci curvature: Open problems (2008), Problem C, https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf; accessed 2026-07-29, with PDF-text extraction recorded in the supplied provenance.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $4000003$, source code catalogue entry, Nilpotent groups. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.