Hydra group doubles are not residually finite

TL;DR

This paper proves that the hydra groups constructed by Dison and Riley, which have extremely fast-growing Dehn functions, are not residually finite, answering a previously open question.

Contribution

It demonstrates that the hydra groups with Ackermann-type Dehn functions are not residually finite, clarifying the limitations of these constructions.

Findings

Hydra groups are not residually finite.

These groups have super-exponential Dehn functions.

Residuall finiteness does not hold for the constructed hydra groups.

Abstract

In 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function $f$, they produce a finitely presented residually finite group with Dehn function dominating $f$. There are no known elementary examples of finitely presented residually finite groups with super-exponential Dehn function. Dison and Riley's hydra groups can be used to construct a sequence of groups for which the Dehn function of the k-th group is equivalent to the k-th Ackermann function. Kharlampovich, Myasnikov, and Sapir asked whether or not these groups are residually finite. We show that these constructions do not produce residually finite groups.