# Polynomial-time proofs that groups are hyperbolic

## TL;DR

This paper introduces a polynomial-time method to analyze finitely presented groups, determining hyperbolicity and producing efficient word problem solvers, leveraging pregroups and van Kampen diagrams.

## Contribution

It presents the first polynomial-time algorithm to verify hyperbolicity of groups and construct linear-time word problem solvers using pregroups and van Kampen diagrams.

## Key findings

- Algorithm often faster than KBMAG
- Successfully identifies hyperbolic groups
- Produces linear-time word problem solvers

## Abstract

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings, to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure which analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and are often many orders of magnitude faster than KBMAG, the only comparable publicly available software.