The bounded cohomology of SL_2 over local fields and S-integers
  It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers. 
Michelle Bucher and Nicolas Monod
math.GR math.GT