We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A ∗C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

The focus of this article is on two forms of isometries and homomorphisms in proximal relator spaces. A practical outcome of this study is the detection of descriptively near, disjoint sets in proximity spaces with application in the study of proximal algebraic structures.

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. 2010 Mathematics Subject Classification: 20F65, 20F10...