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.
Cite this article
Will Dison, Timothy R. Riley, Hydra groups. Comment. Math. Helv. 88 (2013), no. 3, pp. 507–540DOI 10.4171/CMH/294