Curvature and Computation

TL;DR

This paper explores the relationship between negative curvature in geometric group theory and the efficiency of computations within these groups, highlighting a theorem by Gromov that links geometry to computational complexity.

Contribution

It emphasizes the connection between negative curvature and computational efficiency, illustrating how geometric insights can inform group theory and algorithm design.

Findings

Gromov's theorem relates hyperbolic geometry to computational properties of groups

Negative curvature implies certain groups allow efficient algorithms

Applying metric geometry offers new insights into traditional group theory problems

Abstract

When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection between negative curvature and efficient computation. This theorem was one of the clearest early indications that applying a metric perspective to traditional group theory problems might lead to new and important insights.