This question came up in an algebra class I'm teaching. It's not my field and I couldn't find an answer easily, so I thought I would ask it here.

Is the fewest number of relations in a presentation of a group with undecidable word problem known?

The example with the fewest relations I could find by googling was 12 relations in a 1969 paper by Borisov (http://www.ams.org/mathscinet-getitem?mr=0260851) I also found a 1972 paper by Collins (http://www.ams.org/mathscinet-getitem?mr=0314998) which describes a presentation with 12 relations.