Hydra groups

Abstract

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 stringrewriting.

Algorithmically complex residually finite groups

“…The natural homomorphism δ :S(MM 4 ) →S extends to a homomorphismδ :G(MM k ) →Ḡ. By Theorem 4.3 and Lemma 4.13 the images underδ of all elements (14) with w ∈ {w 1 , . .…”

“…One can also try to use the hydra groups [10,14] to construct HNN extensions as above with Dehn functions bigger than any prescribed Ackermann function. The question of whether these groups are residually finite was open when the first version of this paper was written, and is now answered in negative in [49].…”

“…. , w l be the elements from W 0 that appear in the representation of g as a product of elements (14). Let E be the set of words that is equal to one of the w j in S(MM k ).…”

“…Let E be the set of elements ofŜ (i.e., words modulo the commutativity relations) which are equal to one of the w j in S(MM 4 ). Note that every word in E starts with a q-letter by definition of elements (14).…”

“…Suppose that g 1 is not 1. Let T be the subgroup of G(MM k ) generated by elements (14) with (14) with …”