Two weeks ago I gave talk titled “Two Combings of “. The talk was about some material I have been discussing lately with my honours supervisor. The talk went well and I thought it would be worth sharing a written version of what I said.
Geometric Group Theory
Combings are a tool that gets used in a branch of mathematics called geometric group theory. Geometric group theory is a relatively new area of mathematics and is only around 30 years old. The main idea behind geometric group theory is to use tools and ideas from geometry and low dimensional topology to study and understand groups. It turns out that some of the simplest questions one can ask about groups have interesting geometric answers. For instance, the Dehn function of a group gives a natural geometric answer to the solvability of the word problem.
Before we can define what a combing is we’ll need to set up some notation. If is a set then we will write for the set of words written using elements of and inverses of elements of . For instance if , then (here refers to the empty word). If is a word in , we will write for the length of . Thus and so on.
If is a group and is a subset of , then we have a natural map given by:
We will say that generates if the above map is surjective. In this case we will write for when is a word in .
The Word Metric
The geometry in “geometric group theory” often arises when studying how a group acts on different geometric spaces. A group always acts on itself by left multiplication. The following definition adds a geometric aspect to this action. If is a group with generators , then the word metric on with respect to is the function given by
That, is the distance between two group elements is the length of the shortest word in we can use to represent . Equivalently the distance between and is the length of the shortest word we have to append to to produce . This metric is invariant under left-multiplication by (ie for all ). Thus acts on by isometries.
Words are Paths
Now that we are viewing the group as a geometric space, we can also change how we think of words . Such a word can be thought of as discrete path in . That is we can think of as a function from to . This way of thinking of as a discrete path is best illuminated with an example. Suppose we have the word , then
Thus the path is given by taking the first letters of and mapping this word to the group element it represents. With this interpretation of word in in mind we can now define combings.
Let be a group with a finite set of generators . Then a combing of with respect to is a function such that
- For all , (we will write for .
- There exists such that for all with for some , we have that for all .
The first condition says that we can think of as a way of picking a normal form for each . The second condition is a bit more involved. It states that if the group elements are distance 1 from each other in the word metric, then the paths are within distance of each other at any point in time.
Not all groups can be given a combing. Indeed if we have a combing of , then the word problem in is solvable and the Dehn function of is at most exponential. One group that does admit a combing is . This group is generated by and one combing of with respect to this generating set is
The first condition of being a combing is clearly satisfied and the following picture shows that the second condition can be satisfied with .
The discrete Heisenberg group, , can be given by the following presentation
That is, the group has three generators and . The generator commutes with both and . The generators and almost commute but don’t quite as seen in the relation .
Any can be represented uniquely as for . To see why such a representation exists it’s best to consider an example. Suppose that . Then we can use the fact that commutes with and to push all ‘s to the right and we get that . We can then apply the third relation to switch the order of and on the right. This gives us that that . If we apply this relation once more we get that and thus . The procedure used to write in the form can be generalized to any word written using .
The fact the such a representation is unique (that is if , then ) is harder to justify but can be proved by defining an action of on . Thus we can define a function by setting to be the unique word of the form that represents . This map satisfies the first condition of being a combing and has many nice properties. These include that it is easy to check whether or not a word in is equal to for some and there are fast algorithms for putting a word in into its normal form. Unfortunately this map fails to be a combing.
The reason why fails to be a combing can be seen back when we turned into . To move ‘s on the right to the left we had to move past ‘s and produce ‘s in the process. More concretely fix and let and . We have and . The group elements and differ by a generator. Thus, if was a combing we should be able to uniformly bound for all and all .
If we then let , we can recall that
We have that and and thus
The group element cannot be represented as a shorter element of and thus and the map is not a combing.
Can we comb the Heisenberg group?
This leaves us with a question, can we comb the group ? It turns out that we can but the answer actually lies in finding a better combing of . This is because contains the subgroup . Rather than using the normal form , we will use where is a combing of that is more symmetric. The word is defined to be the sequence of ‘s and ‘s that stays closest to the straight line in that joins to (when we view and as representing and respectively). Below is an illustration:
This new function isn’t quite a combing of but it is the next best thing! It is an asynchronous combing. An asynchronous combing is one where we again require that the paths stay close to each other whenever and are close to each other. However we allow the paths and to travel at different speeds. Many of the results that can be proved for combable groups extend to asynchronously combable groups.
Hairdressing in Groups by Sarah Rees is a survey paper that includes lots examples of groups that do or do not admit combings. It also talks about the language complexity of a combing, something I didn’t have time to touch on in my talk.
Combings of Semidirect Products and 3-Manifold Groups by Martin Bridson contains a proof that is asynchornously combable. He actually proves the more general result that any group of the form is asynchronously combable.
Thank you to my supervisor, Tony Licata, for suggesting I give my talk on combing and for all the support he has given me so far.
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