Cayley automatic groups are not Cayley biautomatic
  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. 
Alexei Miasnikov and Zoran Sunic
math.GR