Calibrating word problems of groups via the complexity of equivalence   relations
  (1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations. 
Andr\'e Nies and Andrea Sorbi
math.LO