[1] Despite geometric analogies with general linear groups and mapping class groups, their complexity is generally regarded as more challenging, which has fueled the development of new techniques in the field.
This no longer holds for higher ranks: the Torelli group of
contains the automorphism fixing two basis elements and multiplying the remaining one by the commutator of the two others.
The inner automorphism group itself is the image of the action by conjugation, which has kernel the center
obtained by taking the outer class of the extension of an automorphism of
Therefore, when studying properties that are inherited by subgroups and quotients, the theories of
is the fundamental group of a bouquet of n circles,
can be described topologically as the mapping class group of a bouquet of n circles (in the homotopy category), in analogy to the mapping class group of a closed surface which is isomorphic to the outer automorphism group of the fundamental group of that surface.
Given any finite graph with fundamental group
with one boundary component that retracts onto the graph.
that are in the image of such a map are called geometric.
Such outer classes must leave invariant the cyclic word corresponding to the boundary, hence there are many non-geometric outer classes.
A converse is true under some irreducibility assumptions,[2] providing geometric realization for outer classes fixing a conjugacy class.
Out(Fn) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Fricke-Teichmüller space for a bouquet of circles.
To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3.
A more descriptive view avoiding the homotopy equivalence f is the following.
We will now assign to each remaining edge e a word in
Consider the closed path starting with e and then going back to the origin of e in the maximal tree.
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.
The operation of Out(Fn) on the outer space is defined as follows.
induces a self homotopy equivalence g′ of the bouquet of n circles.
And in the other model it is just application of g and making the resulting word cyclically reduced.
Every point in the outer space determines a unique length function
determines via the chosen homotopy equivalence a closed path in X.
The length of the word is then the minimal length of a path in the free homotopy class of that closed path.
Such a length function is constant on each conjugacy class.
If that edge is a loop it cannot be collapsed without changing the homotopy type of the graph.
So one can think about the outer space as a simplicial complex with some simplices removed.