Shelters in Graphs of Groups

March 26, 2022

Let’s fix a group $G$ that acts cocompactly on a tree $T$ . Forester introduced the notion of a deformation of the tree $T$ and Guirardel and Levitt studied the deformation space $\mathscr{D}$ consisting of trees obtainable from $T$ by a finite sequence of deformations. Usually we assume $G$ to be finitely generated, but it’s not clear to me how often this assumption is necessary. Maybe that’ll be a topic for another blog post.

From Free Splittings to Graphs of Groups

January 6, 2022

In the previous post, we defined free splittings for a group $F$ . The quotient of a free splitting $T$ of $F$ by the action of $F$ is a graph of groups with trivial edge groups. The purpose of this post is to introduce maps of graphs of groups with trivial edge groups and explore lifting maps of the quotient graph of groups to twisted equivariant maps of the tree $T$ .

More on Free Splittings

December 29, 2021

Let $F$ be a group. A free splitting of $F$ is a simplicial tree $T$ equipped with an $F$ -action such that edge stabilizers are trivial. I’ve been thinking a lot about free splittings; here are some of my musings. There isn’t really a punch line here, but since it was clarifying to write this, I may as well post it. We are interested in free splittings where there are finitely many orbits of edges and some vertex has infinite stabilizer.

Git for a Certain Kind of Mathematician

September 28, 2021

As a mathematician, I collaborate with other mathematicians on papers. I’ve used Overleaf, emailing the paper back and forth, and taking turns in Dropbox as my mode of collaboration. I haven’t collaborated using Git, but I sure would like to. The purpose of this post is for mathematicians like for example my younger self, who are familiar with the command line and curious about Git as a tool, and for whatever reason don’t have the patience for something like “Git for Poets.

Boundaries of Free Splittings

July 25, 2021

The other day on Twitter I asked whether it might be the case that a non-proper hyperbolic metric space might still have locally compact Gromov boundary. I was particularly interested in the case of a simplicial tree equipped with the path metric where each edge has length $1$ with some vertices of infinite valence. The answer in this case is no. Instead, the boundary of such a tree is homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$ .

A Nice Coincidence

July 19, 2021

The other day I “proved” the Farrell–Jones conjecture for a family of groups. The word “proved” is in quotes because actually there was nothing to prove, I just happened to be primed to notice that all the conditions were already there. The purpose of this post is to tell you the story of how this came to be and give the sketch of the proof that requires zero input from me.

Gromov Hyperbolicity for Arbitrary Metric Spaces

July 1, 2021

Gromov’s original definition of Gromov hyperbolicity makes sense for arbitrary metric spaces. However, it is only a quasi-isometry invariant for geodesic metric spaces. I learned this from a paper of Väisälä. The purpose of this post is to understand the counterexample he gives. I also define Gromov hyperbolicity and quasi-isometry in this post, which might make it useful for future reference. The reader already familiar with Gromov hyperbolicity and quasi-isometries might wish to skip ahead to the heading below.

The Farey Graph

June 27, 2021

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 $\frac{1}{0}$ .

Free Actions on Real Trees

June 21, 2021

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 $X$ .

Geometric Group Theory Without Boundaries

June 21, 2021

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.