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This is the first in a series of four papers (with research announcementposted on this arXiv) that together develop a decomposition theory forsubgroups of Out(F_n).  In this paper we develop further the theory of geometric EG strata ofrelative train track maps originally introduced in the work of Bestvina,Feighn, and Handel on the Tits alternative, with our focus trained on certain2-dimensional models of such strata called "geometric models" and on theinteresting properties of these models. A secondary purpose of this paper is toserve as a central reference for the whole series regarding basic facts of thegeneral theory for elements of Out(F_n).

Subgroup decomposition in Out(F_n), Part I: Geometric Models