  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.
