Dani Wise has [recently proven a conjecture][1] of Baumslag, that 1-relator groups whose relator is a proper power are residually finite (in fact, linear). The proof makes use of techniques from geometric group theory, in particular using techniques of hyperbolic groups and CAT(0) cube complexes. 


  [1]: http://comet.lehman.cuny.edu/behrstock/cbms/program.html