Topology and Non-Deterministic Polynomial Time Computation : Avoidance of The Misbehaviour of Hub-Free Diagrams and Consequences

TL;DR

This paper explores the relationship between the topology of asymptotic cones of groups and polynomial time computations, introducing conditions to avoid certain diagram misbehaviours linked to hub-free diagrams.

Contribution

It introduces conditions to prevent misbehaviour of hub-free diagrams in the study of groups with small Dehn's function, connecting topology and computational complexity.

Findings

Identifies conditions to avoid misbehaviour of hub-free diagrams.

Establishes links between asymptotic cone topology and polynomial time computations.

Highlights limitations of the proposed method.

Abstract

To study groups with small Dehn's function, Olshanskii and Sapir developed a new invariant of bipartite chords diagrams and applied it to hub-free realization of S-machines. In this paper we consider this new invariant together with groups constructed from S-machines containing the hub relation. The idea is to study the links between the topology of the asymptotic cones and polynomial time computations. Indeed it is known that the topology of such metric space depends on diagrams without hubs that do not correspond to the computations of the considered S-machine. This work gives sufficient conditions that avoid this misbehaviour, but as we shall see the method has a significant drawback.