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Taming the hydra: the word problem and extreme integer compression
Will Dison, Eduard Einstein and Timothy Riley 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. We explain how to compute efficiently with enormous integers represented in highly compressed form using Ackermann functions. This leads to polynomial time solutions to the membership problem for subgroups of huge (Ackermannian) distortion inside hydra groups, and to the word problem for a related family of groups with Ackermannian Dehn function.
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