[Paper Review] Medium-scale curvature at larger radii in finitely generated groups

Robert Kropholler, Brendan Mallery|arXiv (Cornell University)|Apr 22, 2020

Geometric Analysis and Curvature Flows7 references4 citations

TL;DR

This paper introduces a medium-scale Ricci curvature for finitely generated groups using spherical and ball comparison distances at arbitrary radii, showing that dead-end elements yield non-negative curvature and constructing explicit examples of stable positive curvature in the lamplighter and Houghton groups. It demonstrates that this curvature notion differs from Ollivier’s discrete Ricci curvature in non-abelian groups, revealing that optimal transport plans are not always captured by the standard GenCon construction.

ABSTRACT

In this paper we study medium scale curvature, a notion of Ricci curvature for groups. We show that dead-end elements yield non-negative curvature. We make use of related elements to find large classes of positive curvature in the lamplighter and Houghton's group. We also study the effect of increasing the radius of curvature and give examples of positive curvature for arbitrary radius. Finally we make some comparisons between this notion of curvature and the Ricci curvature introduced by Ollivier.

Motivation & Objective

• To extend the notion of Ricci curvature in finitely generated groups beyond radius 1, enabling study of curvature at larger scales.
• To investigate whether positive curvature observed at radius 1 persists when increasing the comparison radius.
• To identify natural classes of group elements—particularly dead-end elements—that yield non-negative curvature across multiple radii.
• To compare the proposed curvature notion with Ollivier’s discrete Ricci curvature, especially in non-abelian groups.
• To explore the role of optimal transport plans in curvature computation and determine when GenCon provides the minimal transport cost.

Proposed method

• Define spherical and ball comparison distances $\mathcal{S}_r(x,y)$ and $\mathcal{B}_r(x,y)$ as the average distance between $xw$ and $yw$ for $w$ in the sphere or ball of radius $r$.
• Introduce curvature via $\kappa_r^\mathcal{S}(x,y) = \frac{d(x,y) - \mathcal{S}_r(x,y)}{d(x,y)}$, with positive curvature indicating contraction under translation.
• Use group symmetry to reduce curvature computation to $\kappa_r^\mathcal{B}(g) = \kappa_r^\mathcal{B}(e, g)$, simplifying analysis via word length and conjugation.
• Analyze dead-end elements with strict depth $k$ to prove $\kappa_r^\mathcal{S}(g) \geq 0$ for all $r < k$, establishing a broad class of non-negative curvature elements.
• Construct explicit examples in the lamplighter group $\mathcal{L}_2$ and Houghton’s group $H_2$ to demonstrate stable positive curvature at arbitrary radii.
• Compare the new curvature with Ollivier’s discrete Ricci curvature by analyzing transport costs and optimal plans, showing discrepancies in non-abelian groups.

Experimental results

Research questions

• RQ1 Do elements with positive curvature at radius 1 remain positively curved when the comparison radius is increased?
• RQ2 Can dead-end elements in finitely generated groups be used to systematically construct non-negative curvature at larger radii?
• RQ3 In which groups does the GenCon construction yield the optimal transport plan for curvature computation?
• RQ4 How does the proposed medium-scale curvature differ from Ollivier’s discrete Ricci curvature in non-abelian groups?
• RQ5 Are there groups for which the optimal transport plan for curvature is independent of the group element, or does it vary significantly?

Key findings

• Dead-end elements of strict depth $k$ exhibit non-negative curvature for all comparison radii $r < k$, providing a general mechanism for constructing such elements.
• The lamplighter group $\mathcal{L}_2$ and Houghton’s group $H_2$ contain elements with positive curvature that persists across all tested radii, demonstrating stability of positive curvature at larger scales.
• In the free group $F_n$, the GenCon construction yields the optimal transport cost, and thus matches Ollivier’s curvature, showing agreement in this case.
• In $S_3$ with standard generators, the GenCon construction does not yield the minimal transport cost, proving that Ollivier’s curvature and the proposed comparison curvature can differ in non-abelian groups.
• The function $\phi_S^r(g)$ mapping group elements to optimal permutations of $B_r$ is not always trivial, and its structure varies with $g$, indicating that optimal transport plans depend on the group element.
• The paper constructs examples of positive curvature at arbitrary radii, showing that positive curvature is not an artifact of small-scale geometry but can be stable at larger scales.