non-smooth turns which do not occur in indivisible fixed elements. The extension
from automorphisms to injective endomorphisms of free groups, although straight-
forward, is new. The concept of inertia is new, and quite useful, and the main
results, Theorems IV.5.4 and IV.5.5, are new.
We thank the Centre de Recerca Matematica of the
Institut d'Estudis Catalans for organizing the Semester on Geometric Methods in
Group Theory, during which some of this article was written.
We are grateful to Mladen Bestvina, Gilbert Levitt, Martin Lustig, and Ted
Turner, for many stimulating conversations, both oral and electronic, which were
of fundamental importance to the completion of this project.
We thank Ferran Ced6 for carefully reading an earlier version, and for suggest-
ing many useful corrections and clarifications.