Anderson's Trick

October 30, 2022

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

Aliasing

October 13, 2022

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.

Another Mathematical Postcard

September 3, 2022

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 $\tilde X - C$ .

Telescoping Surfaces

August 31, 2022

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.

A Coarsely Boundedly Generated Mapping Class Group

August 30, 2022

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 $\operatorname{Map}(\Sigma)$ .

Nonexamples of Coarsely Bounded Generation

August 29, 2022

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.

Tameness and More on Ends

August 27, 2022

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.

Locally Coarsely Bounded Mapping Class Groups

August 17, 2022

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.

Self-similar End Spaces

August 14, 2022

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.

Nondisplaceable Subsurfaces

August 12, 2022

This is the second 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 introduce the main obstruction to coarse boundedness of subsets of a big mapping class group: nondisplaceable finite-type subsurfaces. I also define the mapping class group in this post. The mapping class group Mapping class groups of finite-type surfaces are central objects in modern mathematics, bridging fields as disparate as algebraic geometry and low-dimensional topology.