In order to fight the blog slowdown that seems to have come with the end of the semester, I just want to point out a short paper that caught my attention on the arXiv. Michael Freedman’s paper  defines the width of a group G as follows: Consider all topological spaces whose fundamental group is isomorphic to G and all maps from these spaces to the real line. For each of such map , consider the pre-image in of each interval . The width of the map is the maximum, over all these intervals, of the ranks of the fundamental groups of these pre-images. (Or, more precisely, Freedman takes the rank of the inclusion of each fundamental group into the larger space.) The width of G is the minimum width of all such maps .
Of course, you may have noticed that the title of this post uses the term thin position rather than width. But as the old saying goes, where there’s width, there’s thin position… Well, even if that isn’t an old saying, it’s still generally true. The basic idea behind thin position is to take a width, defined along the lines of Freedman’s definition, but rather than looking at only the maximal level, one looks at all levels via lexicographic ordering. So, given a map as above, consider the ranks of all the non-empty pre-images of intervals (or perhaps consider the ranks of their inclusions into the fundamental group of the whole space). Define the complexity of as the tuple of ranks, with multiplicity, arranged in non-increasing order. (And we may need to allow this to be an infinitely long tuple, but since I’m not claiming any theorems based on this definition, I will leave those details for later or for someone else…)
To compare two complexities for different groups, or for the same group, you apply lexicographic/dictionary ordering: If the first entry in one complexity is higher than the first entry in the other, then the first one is greater. Otherwise, you compare the second entries and so on. The generalized width for a group can then be defined as the minimal complexity over all .
In thin position for 3-manifolds, the numbers in the complexity tuple come from embedded surfaces. When the complexity is minimized, the local minima (the values that are lower than the previous and next values, before you rearrange them to non-increasing order) correspond to incompressible surfaces. So I’m curious what one can say about the inclusions of the fundamental groups of the local minima in the version of thin position that comes from Freedman’s width. But I’ll leave that sort of speculation for another day and finish this post with a speculation in a different vein.
If we want to construct a space with fundamental group G that minimizes the width (or the generalized width), we can start by constructing a 1-dimensional complex from the generators. As Freedman notes, we can choose a for this 1-dimensional complex that cuts it into trees, so the width will be zero. If we then glue in 2-cells for the relations, we will be forced to find a where the pre-images of the intervals are no longer simply connected. In some sense because the definition of width looks at (coarsely) co-dimension one slices, the 2-cells in the complex give us generators in the slices. If we then glue in some 3-cells, they should give us relators in the slices, and higher dimensional cells in the complex should have no effect on the fundamental groups of the slices.
Anyway, this is my intuition so I’d like to ask the question of how accurate this intuition is. If it is accurate, then adding cells above dimension three should not reduce the width of the representative. It also seems like adding in more higher dimensional cells shouldn’t increase the width, so adding all possible ones (as in a classifying space for the group, which has G as its fundamental group and a contractible universal cover) should also minimize the width. I’ll summarize these two ideas as follows:
Question: Is the width of a group always realized by a 3-dimensional cell complex? Can every classifying space realize the width (for the correct )?
I’ll also post this question on the “open questions” page, which seems to have fallen into disuse lately…