Free Groups and Presentations

In the last section we've taken groups, found sets of elements that generate these groups, and constructed a metric graph with a natural isometry corresponding to every group element. We can also take a reverse approach: instead of asking what elements generate a group, we can construct groups by defining them to be generated by some elements.

Free groups

A shift in perspective is in order. Before we looked at groups as either isometries/symmetries of a space or figure or as a bunch of elements with a defined product. Another useful way of looking at groups is as words in some alphabet. For instance, take the alphabet consisting of a single letter: a. Our rule for constructing words is the following. We can use the letter a or its upper case version A. If a word contains the string aA or Aa we can cancel it. Thus, in our language the word aaAa is the same as aa (since we can cancel aA or Aa); we write aaAa=aa.

In the last part of the above activity you should notice that the Cayley graph of the group defined by our one letter language looks exactly the same as the Cayley graph of Z with the generator 1. Both are long "lines". If we simply relabel the letter a in our language as 1 and A as -1, we get the exact same structure as the group of integers Z. If a relabeling of elements produces identical Cayley graphs, we say that the two groups are isomorphic--for our purposes they are the same. Note, however, that just because two Cayley graphs can't be made the same by relabeling does not mean that the groups are not isomorphic. After all, choosing different generators will often lead to two different Cayley graphs of the same group.

We don't have to restrict ourselves to just one letter. We can take, for example the alphabet {a,b}. Again, words are composed of the letters a,A,b,B such that we can cancel aA, Aa, bB, and Bb.

Your finished product from part 2 above should look something like this. This is just a small part of the infinite Cayley graph comprised of the vertices close to the identity. Recall our convention that that each edge of the Cayley graph is length 1. Thus my picture of the Cayley graph is not true to life, so to speak; it's just more convenient to visually shrink the edges when drawing the graph.

Of course we can have arbitrary many letters in our alphabet, with the only cancellation being that of a lower case and an upper case of the same letter. Such groups are called free groups. We'll denote the free group generated by an n letter alphabet F_n. We've already seen that F₁=Z, the group of integers.

Relations

A relation in a group is simply a word that's equal to the empty word. In F₁, for example, some relations are Aa, aAAa, aA, etc. In D₄, relations include L₁L₁, R₉₀R₉₀R₁₈₀, and R₁₈₀L₄L₂. We can use the Cayley graph to simply read off the relations of a group: trace along a path from the identity vertex e to itself, recording the generators corresponding to the edges you pass along the way. Below is part of the Cayley graph of the group Z×Z (or equivalently, F₁×F₁) with generators (1,0) and (0,1).

Activity 4: One word

1. What are some relations in Z×Z? Write them in terms of the generators.
2. Recall that the group F₂ can be generated by letters a and b (with inverses A and B respectively) and consists of all words in these letters up to cancellation of inverses. Suppose we allow the cancellation of another word in addition to the four above, the word abAB. Call this new group H. Show that ab=ba in H.
3. Below is the Cayley graph of F₂. How would you modify it to get the Cayley graph for H? (Hint: some words (vertices) that are distinct in F₂ are equal in H. Which ones? If you're having difficulties, start out by drawing part of the Cayley graph of H around the identity.)

4. How would you relabel the Cayley graph of H to get the the Cayley graph of Z×Z given previously?

As you've see in the activity above, while F₁=Z, we've had to add relations to the group F₂ to get the group Z×Z. These additional relations were all the words we could reduce to the identity by canceling the word abAB, that is, setting abAB=e. A group presentation is a list of letters which generate a group and a list of words which we cancel, usually written as ⟨letters|words⟩. For instance, F₂=⟨a,b| ⟩. The list of canceling words is empty because by convention the words aA, Aa, bB, etc. are implied; they are part of every group presentation since elements in a group must have inverses. As we've seen above, Z×Z=⟨a,b|abAB⟩. Thus a free group is one with no relations other than the implied inverse ones.