Recurrence relations in (s,t)-uniform simplicial complexes

TL;DR

This paper introduces (s,t)-uniform simplicial complexes and explores their geometric properties, including recurrence relations for sphere lengths, asymptotic area ratios, and average Gaussian curvature inside spheres.

Contribution

It defines (s,t)-uniform complexes and analyzes geometric features such as recurrence relations, area growth, and curvature, providing new insights into their structure.

Findings

Sphere lengths follow specific recurrence relations.

The ratio of sphere area to radius converges as radii grow.

Average Gaussian curvature inside spheres is computed.

Abstract

We introduce $(s,t)$-uniform simplicial complexes. We show that the lengths of spheres in minimal filling diagrams associated to loops in such complexes are the terms of certain recurrence relations. We study the limit of the ratio of the area of such spheres over their length as the radii of spheres grow. Besides we compute the average Gaussian curvature for vertices inside these spheres.