For groups given by matrix generators, most questions are undecidable. For groups given by a presentation, many questions are undecidable anyhow:

http://mathoverflow.net/questions/47961/undecidability-in-matrix-groups

http://mathoverflow.net/questions/15957/is-any-interesting-question-about-a-group-g-decidable-from-a-presentation-of-g

http://mathoverflow.net/questions/62339/computing-abelianizations