Groups of intermediate growth
  I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and derives the growth properties from its analytical behaviour. Unfortunately, there are substantial mistakes in the proofs, so the main claims should be considered unsettled. In particular, it is still an open problem whether the growth of the Gupta-Sidki group is intermediate or exponential. 
Laurent Bartholdi