Cayley automatic groups are not Cayley biautomatic

TL;DR

This paper demonstrates that some Cayley automatic groups lack Cayley biautomaticity, and establishes the undecidability of the Conjugacy and Isomorphism Problems within this class, highlighting limitations in their algorithmic properties.

Contribution

It proves the existence of Cayley automatic groups that are not Cayley biautomatic and shows undecidability results for key decision problems in this class.

Findings

Existence of Cayley automatic groups not Cayley biautomatic

Undecidability of the Conjugacy Problem in Cayley automatic groups

Undecidability of the Isomorphism Problem in Cayley automatic groups

Abstract

We show that there are Cayley automatic groups that are not Cayley biautomatic. In addition, we show that there are Cayley automatic groups with undecidable Conjugacy Problem and that the Isomorphism Problem is undecidable in the clas of Cayley automatic groups.