When two free factors A and B of a free group F_n are in general position we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurfac
e projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the proof of the Handel-Mosher hyperbolicity theorem for the splitting complex using (liberal) folding paths.
This is the first in a planned series of papers giving an alternate approach to Zlil Selas work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a f
initely generated group and F is a non-abelian free group.
We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.
