Hydra groups

TL;DR

This paper constructs examples of complex groups with highly distorted free subgroups and extravagant Dehn functions, illustrating intricate geometric and algebraic properties through string-rewriting analogies.

Contribution

It introduces new examples of groups with extreme distortion and Dehn functions, linking geometric group theory with combinatorial string-rewriting phenomena.

Findings

Existence of CAT(0), biautomatic, free-by-cyclic groups with huge distortion

Groups with extremely large Dehn functions

Connection between group properties and Hercules-versus-the-hydra string battles

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 string-rewriting.