Perhaps the fundamental observation in geometric group theory, the Milnor–Schwarz lemma (an independent discovery of both Milnor and Schwarz, sometimes romanized Švarc, hence sometimes titled with the names transposed) says, very roughly, that if you have a group of symmetries of a geometric object that satisfies certain properties, then the group of symmetries may be regarded as a geometric object in its own right and if you “squint,” the geometry of the group and the object are the same.

This week I had the pleasure of attending a seminar talk Nick Vlamis gave at CUNY, where he taught us a very pretty trick due to Anderson which one can use, as Anderson did, to prove that the groups of orientation-preserving homeomorphisms of the $2$ - and $3$ -spheres are simple. The purpose of this post is to reproduce Nick’s exposition of the trick. Suppose $g$ is a homeomorphism of a surface $S$ (for me a surface is a Hausdorff and second countable $2$ -manifold) whose support

Perhaps my main non-mathematical joy is music. I took piano lessons from when I was eight through the end of college, and this year I wrote and recorded an album of pop songs, about which more certainly sometime soon. Anyway, I lost most of my weekend to a musical math problem. The purpose of this post is to explain the problem and the solution, partly so that I will have fewer steps to retrace, should I need to in the future.

I submitted another “mathematical postcard” (see below) to the Nearly Carbon Neutral Geometric Topology conference. The purpose of this post is to offer a little more context and explanation for it. For some reason I end up discussing things like ends of groups and finiteness properties rather than sticking strictly to Bieri–Eckmann duality. Here is the postcard: The text reads as follows: A space $\tilde X$ is $n$ -connected at infinity if for all $K \subset \tilde X$ compact, there exists $C \subset K$ compact such that every map $S^n \to \tilde X - K$ is nullhomotopic in $\tilde X - C$ .

this is the eigth and final post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. The purpose of this post is to tie up a loose end. In the third post, we observed that if $\operatorname{Map}(\Sigma)$ is to be coarsely bounded, then any finite, $\operatorname{Map}(\Sigma)$ -invariant set of ends has cardinality at most two, but then went on to discuss the case of self-similarity, where such a set has cardinality one.

This is the seventh post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. The purpose of this post is to show that the following surface from the first post has mapping class group generated by a coarsely bounded set. We showed in the fourth post that $\Sigma$ has locally coarsely bounded mapping class group, hence by results of the previous post it suffices to show that $U_S$ for $S$ some finite-type subsurface, together with a finite set of elements $\phi_i$ , generates $\operatorname{Map}(\Sigma)$ .

This is the sixth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. The purpose of this post is to discuss when a Polish group is generated by a coarsely bounded set, and give examples of mapping class groups which are locally coarsely bounded but fail this criterion. The general case We have the following theorem of Rosendal. Theorem 1.2 (Rosendal). Let $G$ be a Polish group.

This is the fifth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. At the beginning, I mentioned that Mann and Rafi provide a classification of which $\Sigma$ have $\operatorname{Map}(\Sigma)$ generated by a coarsely bounded set under a hypothesis. The purpose of this post is to explore that hypothesis, which is called tameness. Additionally, I want to report on recent progress by Mann and Rafi in their note Two results on end spaces of infinite-type surfaces.

This is the fourth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. Truth be told, I’ve made many abortive attempts at writing this post. I want to describe the classification of locally coarsely bounded mapping class groups in a way that is useful and not too technical. I’m going to try and talk through two examples that Mann–Rafi give at the beginning of the paper, and then state the classification theorem at the end of the section.

This is the third post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. In the previous post we explored an obstruction to coarse boundedness, while in this one we will introduce a sufficient condition for $\operatorname{Map}(\Sigma)$ to be coarsely bounded. This post includes proofs that closely follow those in the paper, which while elementary, get a little involved. I imagine this will be one of very few moments in the series where I give full proofs.