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

Taming the hydra: the word problem and extreme integer compression