I’m sitting in my bathrobe on the couch eating a bowl of chicken soup while husband watches baby, which is all to say that I apologize if this fever-tinged post makes less sense/is less factual than my usual math posts.
This post is the story of my fairly young field of math, informed by the folklore I’ve heard in the three years since I first heard the words “geometric group theory” and some wikipedia.
Pure math is, very very roughly, divided into three main areas: algebra, analysis, and topology. There’s a whole bunch of other math that doesn’t fit in here (logic, set theory, category theory off the top of my head), but these are the three required core courses that every U.S. Ph.D. student studies in their first one or two years of grad school. Geometric group theory lives between algebra and topology- “group theory” is the study of groups, which we’ve seen a few times before, and “geometric” means that we’ll be looking at shapes. Geometric group theory (GGT for short) uses geometric/topological methods and ideas to come to conclusions about groups associated with shapes.
There are a few main ways to associate groups to shapes: the first we learn is the fundamental group, which will get its own post sometime- this group records different loops on our topological shape. There’s also homology groups, cohomology, mapping class group, higher homotopy groups, etc. etc. These all record different info about the shape. The fundamental group of a circle is the integers, of a torus is , or pairs of integers, and of the n-holed torus is , as in the picture above.
Speaking of segues, geometric group theory started as a way to answer some questions (as fields of math are wont to do). In the 1910s, a mathematician named Max Dehn posed three questions about groups. To understand them we’ll need to know about group presentations, which is just a standard (but not canonical) way to write groups. So take your group, and look at the generators you have. Label each generator by a letter in an alphabet- we have a good one, it starts with “a” and moves on to “b” but you could also do if you wanted. Then write down all the true equations involving your generators. This is best done with an example also I am done with my soup =(
Let’s take the group of pairs of integers, . We’ll use (0,1) and (1,0) as our generators, since any pair (x,y) can be written as x(1,0) + y(0,1). Let’s label them by a=(1,0) and b=(0,1). Then a true equation is a+b=b+a. Since we’re in group-land, let’s skip the “+” sign and say addition is our group operation, and write ab=ba, or equivalently, , where I used e for the identity element (0,0). Then our group presentation is .
Back to Dehn’s problems!
One was the word problem: given a word in your alphabet, could you tell if it was the identity element? It’s clear (0,1)+(1,0)-(0,1)-(1,0)=(0,0), but what about a word like in our group presentation? Actually the question of whether a word is trivial (another way of saying equal to the identity) in our group presentation is pretty easy to answer: just count up the exponents of each letter. If they sum to 0, then you’re trivial. But in general this is hard. Dehn’s other two problems were also hard.
Group theorists used combinatorial (roughly, counting) methods to try to answer Dehn’s problems, and wrote algorithms (Dehn did this, actually) to tackle them. On the way they built up combinatorial group theory, and drew lots of pretty pictures of trees (graph theory) and planar diagrams (which is what I do a lot of). According to wikipedia, in the 1980s GGT started appearing after Gromov wrote a thing on hyperbolic groups (hyperbolic post here). That was around the time that people started realizing that you could generalize properties of groups: instead of saying oh, group A has properties 1,2,3, and so do groups B, C, D, etc., you could say all groups that are quasi-isometric to group A have properties 1, 2, 3. Understanding groups up to quasi-isometry is one of the main goals of geometric group theory. (quasi-isometry defined in this post).
And now geometric group theory is a thriving young field. You can tell from this page that UIC is one of the big GGT departments in the world, and this page shows all the conferences going on about it. In fact, I’d say all GGT mathematicians on the internet know Jon McCammond’s GGT website.
And that’s all I have to say about what geometric group theory is. Back to very important work, napping. Apologies again for lack of clarity, precision, sense-making…