In an earlier post we introduced the concept of an ideal edge in a graph of groups and described how to blow up ideal edges to construct new graphs of groups. To prove that (the reduced spine of) Outer Space for a virtually free group is contractible, Krstić and Vogtmann define what is essentially a combinatorial analogue of a non-singular Morse function on Outer Space and show that for a well-chosen function, the set of minimal vertices of the complex is actually the star of a single reduced marked graph of groups, which is contractible.

Martin Pettet characterized which virtually free groups have finite outer automorphism group. Krstić and Vogtmann studied what I’ll call the “spine of reduced Outer Space” for a virtually free group and gave a formula for computing its dimension. I want to point out that the dimension of this complex can be arbitrarily big even when the virtually free group has finite outer automorphism group, so this post will be given to understanding a particular example.

Let’s once again fix a group $G$ that acts cocompactly on a tree $T$ . In the previous post we defined what it meant for an edge of $T$ to be surviving or collapsible. Collapsing orbits of collapsible edges in $T$ yields a new tree $T'$ which we say is obtained from $T$ by forest collapse. Two trees $T$ and $T'$ are in the same deformation space if there are $G$ -equivariant maps $T \to T'$ and $T' \to T$ .

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.

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 $T$ .

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.

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.

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 $\mathbb{N}^\mathbb{N}$ .

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’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.