Rank 3 Outer Space

Rank 2 Outer Space

Image from "On the Geometry of Outer Space" by Karen Vogtmann

Given a finite graph, the fundamental group is some free group, F_n. The rank of this free group (n) is called the rank of the graph. For example, the rank 2 graphs are those which have two distinct 'loops'.

Rank n "outer space" is the space of 'marked' metric graphs of rank n. To avoid what are essentially duplicates, we impose that the sum of the lengths of the edges is equal to 1. We also require the valence of every vertex be at least 3. The motivation for building such a space, is that the "outer automorphism group of F_n" act on this space in a nice way.

The open simplices in rank 2 outer space are families of graphs of one of three types: theta, figure eight or barbell. Theta and barbell simplices have dimension two, since the three edges can take different lengths, with the restriction their sum is 1. On the other hand, figure eight cells have dimension 1, since they only have two edges.

The three figure eight edges on a theta cell correspond to the result of shrinking each of the three edges. There is only one edge of the barbell we can shrink to obtain a figure eight graph. Shrinking the other two leads to a degenerate graph which no longer has rank 2.

Our goal here is to understand the types of graphs in each open simplex in rank 3 outer space, the dimension of the simplex, and how these simplices bound each other. It is far to large in dimension to visualize the entire space as nicely as we can in rank 2.

Dimension 2

Let's start with the simplest rank 3 graph: the rose with three petals. Open simplices with graphs of this type are two dimensional. Since collapsing any of the three edges results in a degenerate graph, none of the boundary of such a simplex is part of rank 3 outer space.

A useful tool as we increase dimension is the sub-tree = "a maximal sub graph which is a tree". In dimension 2, this sub-tree could only be a vertex. In general, the number of edges on a subtree is the total number of edges minus the rank. So in our context, a graph in a dimension n open simplex has a subtree with n - 2 edges.

Dimension 3

Graphs in dimension 3 open simplices have a subtree consisting of one edge. From here, we need to consider all ways to attach three more edges which result in a rank 3 graph. We could attach all three edges as loops at one end of the edge, but then the other end has a vertex with valence 1. We can also attach an edge on either side of the subtree. By considering all valid ways to combine these two ways of adding in edges, we get the four types of graphs below. The subtree is green, while the three new edges are blue.

Dimension 4

Graphs in simplices with dimension 4 have a subtree consisting of two edges. Following a similar process to above, we can flesh out all possible valid ways to add three new edges and get a valid rank 3 graph.

Dimension 5

Graphs in simplices with dimension 5 have a subtree with three edges. There are two such trees: edges all in a line, or in a Y shape. Following the same process, we can find all types of graphs with either subtree.

More Dimensions?

Suppose there is an open simplex which is 6 dimensional in rank 3 outer space. A graph in such an open simplex will be rank 3 with 7 edges, and hence have a subtree with 4 edges. The subtree has at least 5 vertices and the sum of the valence (within the subtree) of all the vertices is 8. After adding in 3 new edges, the total valence sum is 8+6 = 14. However, in order to be a valid graph, each of the 5 vertices must have valence at least 3, meaning the valence sum must be at least 15. Hence the subtree can have at most 3 edges, so an open simplex in rank 3 outer space can be at most 5 dimensional.

How do these simplices glue together?

Consider any of the dimension 3, 4, or 5 graph types. Choose any edge, and shrink it to a point. The resulting graph may not be rank 3, or may have a valence 2 vertex. If this happens, we approached a boundary of a simplex which does not belong to rank 3 outer space. However, depending on choice of edge, the resulting graph may be a valid rank 3 graph which lives in an open simplex of one dimension lower. In this case, we approached a boundary of the simplex which is part of rank 3 outer space, and we can find this graph type in our list!

Here we have the complete diagram of all simplex types, and how they bound each other. The number of arrow heads indicates the number of boundary simplices which have that type. "Arrows" with circles indicate boundaries which are not part of rank 3 outer space.

The green shaded graphs on the left are not part of "reduced" outer space. In other words, these graphs all contain an edge which, when removed, results in a disconnected graph. On the other hand, the cells on the right have no such edge.

Rank 4 Reduced Outer Space

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