The group Aut(Fn ) of automorphisms of a finitely generated free group acts properly and cocompactly on a simply-connected simplicial complex known as the degree 2 subcomplex of the spine of Auter space. In the first part of this thesis, we show that the degree 2 subcomplex contains a proper, invariant, simply-connected subcomplex Kappa, and use Kappa to simplify a finite presentation of Aut(Fn ) given by Armstrong, Forrest, and Vogtmann. Further, we prove that Kappa is contained in every Aut(Fn )-invariant simply-connected subcomplex of the degree 2 subcomplex. The mapping class group MCG+/- (sub upsilon)x(epsilon) of an orientable, basepointed, punctured surface (epsilon, upsilon) acts properly and cocompactly on a simplicial complex known as R(sub (epsilon, upsilon) the ribbon graph complex of (epsilon, upsilon). We define a filtration Jsub0 C Jsub1 C Jsub2 . . . on R(sub epsilon,upsilon) and prove that Jsubi is i-dimensional and (i - 1)-connected.