  We introduce and study the notions of boundary actions and of the Furstenberg
boundary of a discrete quantum group. As for classical groups, properties of
boundary actions turn out to encode significant properties of the operator
algebras associated with the discrete quantum group in question; for example we
prove that if the action on the Furstenberg boundary is faithful, the quantum
group C*-algebra admits at most one KMS-state for the scaling automorphism
group. To obtain these results we develop a version of Hamana's theory of
injective envelopes for quantum group actions, and establish several facts on
relative amenability for quantum subgroups. We then show that the Gromov
boundary actions of free orthogonal quantum groups, as studied by Vaes and
Vergnioux, are also boundary actions in our sense; we obtain this by proving
that these actions admit unique stationary states. Moreover, we prove these
actions are faithful, hence conclude a new unique KMS-state property in the
general case, and a new proof of unique trace property when restricted to the
unimodular case. We prove equivalence of simplicity of the crossed products of
all boundary actions of a given discrete quantum group, and use it to obtain a
new simplicity result for the crossed product of the Gromov boundary actions of
free orthogonal quantum groups.
