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.
Fundamental group of this four holed surface is quadruples of the integers (the mouth is a hole but not the eyes)
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 
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 
Let’s take the group of pairs of integers, 
Back to Dehn’s problems!
Word, dog.
But actually you pronounce Dehn like a great Dane. I don’t mean to say the mathematician was a dog, just that his name sounds like a dog. This was funnier before I started writing the caption.
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 
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…
Baking and Math 
Quick question, how did you insert latex into your blog? Nice post by the way.
Thank you! And thanks for stopping by. You add $ signs around whatever you want with a “latex” written after the first dollar sign, so for instance $%latex \mathbb{Z}$ without the % sign will give you
That’s so intuitive! I will be using that from now on. Thank so much and keep writing more math posts, there isn’t much higher level math on these blogs.
Hi, I just wanted to point out for future readers that the fundamental group of a surface of positive genus is not generally free abelian. Such groups admit presentation $\langle a_1,…,a_{2g}~|~\prod_{i=1}^{g-1}[a_i,a_{i+1}]\rangle$, which is abelian only when $g=1$.