  We extend Lang's conjectures to the setting of intermediate hyperbolicity and
prove two new results motivated by these conjectures. More precisely, we first
extend the notion of algebraic hyperbolicity (originally introduced by
Demailly) to the setting of intermediate hyperbolicity and show that this
property holds if the appropriate exterior power of the cotangent bundle is
ample. Then, we prove that this intermediate algebraic hyperbolicity implies
the finiteness of the group of birational automorphisms and of the set of
surjective maps from a given projective variety. Our work answers the algebraic
analogue of a question of Kobayashi on analytic hyperbolicity.
