Brody hyperbolicity and homotopy

The paper's aim is to develop a theory in which the concept of Brody hyperbolicity of a complex space (cfr. [2]) is interpreted in terms of homotopy-theoretic structures. We contend that this interplay will be particularly useful if implemented by applying homotopy-theoretical techniques and constructions to get information on hyperbolic spaces. Imitating the construction of homotopy groups, we will define holotopy groups that will be able to tell apart different complex structures. From our point of view, the most important feature of these groups is that they vanish in a certain range if evaluated on a Brody hyperbolic complex space (see Theorem 4.1), providing therefore a way to reduce the proof of non hyperbolicity of a complex space to the existence of a nonzero holotopy class in these groups.