Homological methods in rigidity theory using graphs of groups

TL;DR

This paper develops algebraic tools using cellular sheaves to analyze infinitesimal rigidity in graph-of-groups models, generalizing known results in rigidity theory.

Contribution

It introduces an algebraic framework for studying infinitesimal rigidity via cellular sheaves and extends Maxwell-count criteria to Lie group settings.

Findings

Henneberg moves preserve independence under certain conditions.

Infinitesimal rigidity and flexibility are shown to be generic properties.

Maxwell-count provides a necessary and sufficient condition for minimal rigidity in specific Lie group contexts.

Abstract

In recent work, Stokes and Vermant considered graph-of-groups realisations of hypergraphs as a new description of rigidity-theoretic problems. In this paper, we show that the infinitesimal aspects of graph-of-groups realisations can be analysed using cellular sheaves and their cohomology. Using these tools, we give an algebraic condition for Henneberg moves to preserve independence, and we prove that the infinitesimal rigidity and flexibility of certain graph-of-groups realisations are generic properties. We use these results to show that whenever a rigidity-theoretic problem is defined in a Lie group $G$ using a $1$-dimensional connected subgroup $H$ with $N_{G}(H)/H$ finite, then the so-called Maxwell-count leads to a necessary and sufficient condition for minimal rigidity, generalising various known results in the literature.