Finitely presented subgroups of automatic groups and their isoperimetric   functions
  We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality. 
Gilbert Baumslag, Martin Bridson, Charles Miller and Hamish Short