Simple loop conjecture for limit groups
  There are noninjective maps from surface groups to limit groups that don't kill any simple closed curves. As a corollary, there are noninjective all-loxodromic representations of surface groups to SL(2,C) that don't kill any simple closed curves, answering a question of Minsky. There are also examples, for any k, of noninjective all-loxodromic representations of surface groups killing no curves with self intersection number at most k. 
Larsen Louder
math.GR