High Quality Content by WIKIPEDIA articles! In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory. A point of the outer space is essentially an R-Graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from X to the Bouquet of n circles. An R- Graph is just a weighted Graph with weights in R. The sum of all weights should be 1 and all weights should be positive. To avoid disambiguity (and to get a finite dimensional space) it is furthermore required, that the valency of each vertex should be at least 2. A more descriptive view avoiding the homotopy equivalence f is the following. We may fix an identification of the fundamental group of the bouquet of n circles with the free group Fn in n variables. Furthermore we may choose a maximal tree in X and choose for each remaining edge a direction. We will now assign to each remaining edge e a word in Fn in the following way.

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