Out(Fn)
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.
Out(Fn) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.
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 ambiguity (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. Consider the closed path starting with e and then going back to the origin of e in the maximal tree. Composing this path with f we get a closed path in a bouquet of n circles and hence an element in its fundamental group Fn. This element is not well defined; if we change f by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique cyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type of f from these data. This view has the advantage, that it avoids the extra choice of f and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.
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