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Hydra groups

Will Dison and Tim Riley

Commentarii Mathematici Helvetici, 88(3), pages 507–540, 2013

We give examples of CAT(0), biautomatic, free–by–cyclic, one–relator groups which have finite–rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules–versus–the–hydra battles in string–rewriting.