  Let S be any orientable surface of infinite genus with a finite number of
boundary components. In this work we consider the curve complex C(S), the
nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the
topological ends of S carry genus, we show that all elements in the
automorphism groups Aut(C(S)), Aut(N(S)) and Aut(G(S)) are geometric, i.e.
these groups are naturally isomorphic to the extended mapping class group
MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within
Aut(C(S)) and Aut(N(S))
