  Gapless edge modes hosted by chirally-stacked trilayer graphene display
unique features when a bulk gap is opened by applying an interlayer potential
difference. We show that trilayer graphene with half-integer valley Hall
conductivity leads to unbalanced edge modes at opposite zigzag boundaries,
resulting in a natural valley current polarizer. This unusual characteristic is
preserved in the presence of Rashba spin-orbit coupling that turns a gated
trilayer graphene into a ${Z}_2$ topological insulator with an odd number of
helical edge mode pairs.
