The Farey graph, or Farey diagram, is an object that appears in many guises throughout math. For me, it appears several times as a complex related to the outer automorphism group of a free group of rank two, but it has connections to things like continued fractions as well. The purpose of this post is introduce the Farey graph and prove a couple of basic properties of it. The Farey graph has as vertices all rational numbers expressed as fractions $\frac{p}{q}$ in lowest terms with $q \ge 0$ , together with a vertex for $\frac{1}{0}$ .

A famous theorem of Rips says that a finitely generated group with a free action on a real tree is a free product of free abelian and surface groups. The purpose of this blog post is to introduce real trees and illustrate this theorem by constructing free actions of such groups on real trees. Let us begin by defining real trees. An arc in a space $X$ is an embedding $\gamma\colon [a,b] \to X$ of a closed interval of $\mathbb{R}$ in $X$ .

Along with Macarena Arenas, Sami Douba and Abdul Zalloum, I am organizing a virtual “summer school” that began today and will run until July 2. More info is available on the summer school website. This is the second iteration of the summer school! I spoke in the first iteration; the purpose of this post is to house links to the notes, recordings and exercises for the mini-course I gave. I gave three lectures in this summer school on Bass and Serre’s theory of graphs of groups and groups acting on trees.

Sometimes a pullback square is an intersection! Here’s a short post about something cute that came up in my research this week. Aside from NCNGT, which has been very lovely, my math this week has been carefully walking through a small side proposition in a paper I’m working on with Ty Ghaswala. The idea is that graphs of groups come with a notion of a covering space due to Bass, and étale groupoids come with their own notion of covering spaces (for us, we’re following the definition in Bridson–Haefliger), and given that you can construct an étale groupoid associated to any graph of groups, I wanted to reconcile the two notions.

I’m giving a talk in Nearly Carbon Neutral Geometric Topology about Nielsen realization for infinite-type surfaces. The format for the conference is that everyone records and uploads their talks beforehand, and then the conference format allows for discussion and comments via Discord. The purpose of this post is to share and talk a little bit about my talk, which is below. The Nielsen realization problem was posed by Nielsen in 1932.

I submitted a “mathematical postcard” (see below) to the Nearly Carbon Neutral Geometric Topology conference. The purpose of this post is to share the postcard and offer a little more context and explanation for it. I’m also giving a talk in said conference, but I’ll talk about that in a separate post. Here is the postcard: Let me say a little more about the space we’re talking about. Let $F = A_1*\dotsb*A_n*F_k$ be a free product of the groups $A_1,\dotsc,A_n$ with a free group of rank $k$ .

The purpose of this post is to share some talk notes I made on orbifolds and graphs of groups as étale groupoids. The talk comes out of joint work in progress with Tyrone Ghaswala. It starts with us realizing that some representations of the braid group into the automorphism group of a free group that each of us knew how to obtain by different methods were really the same representation. We thought that there ought to be some way to reconcile these two representations topologically by working with the mapping class group of certain 2-dimensional orbifolds.

Train track maps, which also come in “relative” and other flavors, are particularly nice homotopy equivalences of graphs used as tools to reason about outer automorphisms of free groups. I wrote a paper extending train track maps (as well as relative train track maps and “CTs”) to graphs of groups (with trivial edge groups for CTs). In this post I want to talk a little about how to think about train track maps and (“tame” in some sense) homotopy equivalences of graphs of groups more generally.

This is a post about one of those things that is really a case of category theory cheekily playing with unhooking a signifier from what it typically signifies. Namely, one of the (many) tricky foundational things about working with orbifolds goes by the name “the axiom of choice is false for the category of smooth manifolds.” One way to misinterpret the statement is to take it literally: as undergraduate math majors learn, the axiom of choice says that given an infinite collection of sets, you can always choose exactly one element of each set, even if the collection is infinite.

Given a topological space $X$ and a continuous map $\pi\colon X \to Y$ , there is an adjunction between the categories of sheaves on $X$ and sheaves on $Y$ . The adjunction is somewhat mysterious to me as I begin to write this, so the hope is that by writing about it, I’ll begin to understand. The functors in question are the pushforward and inverse image functors, which arise naturally in algebraic geometry (so I am led to believe).