Taming the hydra: the word problem and extreme integer compression

TL;DR

This paper demonstrates that despite groups having extremely fast-growing Dehn functions due to the hydra phenomenon, their word problems can still be solved efficiently using novel methods for handling compressed representations of enormous integers.

Contribution

It introduces a new approach for efficiently computing with large integers represented by Ackermann function strings, enabling polynomial-time solutions to complex word problems.

Findings

Polynomial-time solutions for groups with Ackermannian Dehn functions

Efficient algorithms for compressed large integer computations

Novel use of Ackermann functions for group theory problems

Abstract

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison & Riley showed that a "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.