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The object of this expository work is to try to unveil thetopological/geometric intuition behind the theory of free groups and theirautomorphism and outer automorphism groups. The method we follow is to focus ona series of problems in the study of free groups, and use the solutions ofthose problems to motivate topological/geometric tools. We do not aim to writedown proofs which minimize the number of alphanumeric characters. We insteadstrive to write down proofs which maximize the development of broadlyapplicable geometric tools.  In Part I we study problems solved by Nielsen and Whitehead in the 1920's and1930's, but we approach these problems from a modern topological/geometricviewpoint, and we formulate their solutions so as to motivate modern tools,including marked graphs, the outer space of a free group, and fold paths inouter space.

The topology, geometry and dynamics of free groups. Part I: Outer space, fold paths, and the Nielsen/Whitehead problems