We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as “medium-scale” because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature [Formula: see text], abelian groups are identically flat, and in the other direction we show that [Formula: see text] implies the group is virtually abelian. Beyond that, [Formula: see text] captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study.

No abstract

In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. Empirically, we demonstrate that the effect of using the Cartesian latitude-longitude projection on the order of Reock scores is quite dramatic.

No abstract

Let D n be the n-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most n+1 3 . On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.