Pursued by Stacks 1: CFGs

May 16, 2022

I’ve spent the past couple years on and off understanding how to think of graphs of groups and orbifolds as (being represented by) étale groupoids. In view of Lerman’s critique, to which his answer is the category of geometric stacks over manifolds, I thought it might be relevant to try and understand what a stack is. This is the first in what will likely be several posts towards understanding the definition of a stack; this one is focused on the concept of a category fibered in groupoids.

More Maps: Lerman's Critique

May 15, 2022

Lerman critiques the category of étale groupoids with arrows generalized maps by arguing that there is a problem with gluing maps. The purpose of this blog post is to try to digest this critique. We follow the notation set up in the previous two blog posts. Lerman’s critique takes place in the realm of bibundles, but as we saw in the previous post, it’s possible to translate from the bibundles point of view to the generalized maps point of view.

Maps of Étale Groupoids

May 15, 2022

The purpose of this post is to explore two or three notions of a map between étale groupoids. We follow the notation for étale groupoids established in the last post. Functors If $\mathcal{G}$ and $\mathcal{H}$ are étale groupoids, which are in particular categories, a natural class of maps of étale groupoids to consider would be the continuous functors. A functor $f\colon \mathcal{G} \to \mathcal{H}$ is a pair of maps $f_0\colon \mathcal{G}_0 \to \mathcal{H}_0$ and $f_1\colon \mathcal{G}_1 \to \mathcal{H}_1$ that commute with the various structure maps.

Graphs of Groups as Étale Groupoids

May 14, 2022

It’s been a while since I talked about étale groupoids! Time to rectify this situation. The purpose of this post is to explain how to see graphs of groups as étale groupoids. In a later post I’d like to explore the notion of maps between étale groupoids. Groupoids A groupoid is a small category $\mathcal{G}$ in which all arrows are invertible. Thus there are two sets, $\mathcal{G}_0$ and $\mathcal{G}_1$ of objects and arrows respectively, and several structure maps:

A Cut Vertex Lemma

May 10, 2022

Building on ideas from the previous post regarding the star graph, I wrote a short paper generalizing a classical lemma of Whitehead. The purpose of this post is to describe Whitehead’s lemma and my generalization. In the previous post, we met the star graph associated to a collection of conjugacy classes of elements of a virtually free group. It should be reasonably clear that one can extend the notion of a star graph to conjugacy classes of elements of a group that acts cocompactly on a tree.

The Star Graph

April 2, 2022

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.

A Virtually Free Observation

March 29, 2022

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.

Ideal Edges

March 27, 2022

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 $T' \to T$ .

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 $T$ .