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It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last post, because I wanted to get to the bottom of a few analytic details in the notorious Gromov-Schoen paper. I am not quite at the bottom yet, but maybe closer than I was; but I’m still pretty far from having collected my thoughts to the point where I can do them justice in a post. So I’ve finally decided to put Kähler groups on the back burner for now, and resume my usual very sporadic blogging habits.

So the purpose of this blog post is to advertise that I wrote a little piece of software called kleinian which uses the GLUT tools to visualize Kleinian groups (or, more accurately, interesting hyperbolic polyhedra invariant under such groups). The software can be downloaded from my github repository at

https://github.com/dannycalegari/kleinian

and then compiled from the command line with “make”. It should work out of the box on OS X; Alden Walker tells me he has successfully gotten it to compile on (Ubuntu) Linux, which required tinkering with the makefile a bit, and installing freeglut3-dev. There is a manual on the github page with a detailed description of file formats and so on.

The purpose of this brief blog post is to advertise that I wrote a little piece of software called wireframe which can be used to quickly and easily produce .eps figures of surface for inclusion in papers. The main use is that one can specify a graph in an ASCII file, and the program will then render a nice 3d picture of a surface obtained as the boundary of a tubular neighborhood of the graph. The software can be downloaded from my github repository at

https://github.com/dannycalegari/wireframe

and then compiled on any unix machine running X-windows (e.g. linux, mac OSX) with “make”.

The program is quite rudimentary, but I believe it should be useful even in its current state. Users are strenuously encouraged to tinker with it, modify it, improve it, etc. If you use the program and find it useful (or not), please let me know.

A couple of examples of output (which can be created in about 5 minutes) are:

and

(added Feb. 20, 2013): I couldn’t resist; here’s another example:

(update April 12, 2013:) Scott Taylor used wireframe to produce a nice figure of a handlebody (in 3-space) having the Kinoshita graph as a spine. He kindly let me post his figure here, as an example. Thanks Scott!

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which are more natural and easier to understand.

If $G$ is a group, and $a,b$ are elements of $G$, the commutator of $a$ and $b$ (denoted $[a,b]$) is the expression $aba^{-1}b^{-1}$ (note: algebraists tend to use the convention that $[a,b]=a^{-1}b^{-1}ab$ instead). Commutators (as their name suggests) measure the failure of a pair of elements to commute, in the sense that $ab=[a,b]ba$. Since $[a,b]^c = [a^c,b^c]$, the property of being a commutator is invariant under conjugation (here the superscript $c$ means conjugation by $c$; i.e. $a^c:=cac^{-1}$; again, the algebraists use the opposite convention).