Latest Posts

A Polish Milnor–Schwarz Lemma
November 6, 2022
Perhaps the fundamental observation in geometric group theory, the Milnor–Schwarz lemma (an independent discovery of both Milnor and Schwarz, sometimes romanized Švarc, hence sometimes titled with the names transposed) says, very roughly, that if you have a group of symmetries of a geometric object that satisfies certain properties, then the group of symmetries may be regarded as a geometric object in its own right and if you “squint,” the geometry of the group and the object are the same.

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 orientationpreserving homeomorphisms of the \(2\) and \(3\)spheres are simple. The purpose of this post is to reproduce Nick’s exposition of the trick.

Aliasing
October 13, 2022
Perhaps my main nonmathematical 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.

Telescoping Surfaces
August 31, 2022
This is the eighth and final post in a series on Katie Mann and Kasra Rafi’s paper Largescale 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 selfsimilarity, where such a set has cardinality one. I’d like to discuss that remaining case.

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 Largescale 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.

Nonexamples Of Coarsely Bounded Generation
August 29, 2022
This is the sixth post in a series on Katie Mann and Kasra Rafi’s paper Largescale 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.

Tameness And More On Ends
August 27, 2022
This is the fifth post in a series on Katie Mann and Kasra Rafi’s paper Largescale 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 infinitetype 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 Largescale 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.

SelfSimilar End Spaces
August 14, 2022
This is the third post in a series on Katie Mann and Kasra Rafi’s paper Largescale 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 Largescale 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 finitetype subsurfaces. I also define the mapping class group in this post.

Ends Of Surfaces
August 11, 2022
This is the first post in a series on Katie Mann and Kasra Rafi’s paper “Largescale geometry of big mapping class groups.” The purpose of this post is to introduce the classification of (possibly) infinitetype surfaces by their genus and end spaces, and to introduce Mann and Rafi’s preorder on the end space.

Groups As Metric Spaces
August 10, 2022
A week ago I attended Big Mapping Class Groups at BC, a workshop where participants, myself included, presented on papers about mapping class groups of infinitetype surfaces that we did not write. I’d like to spend a couple of posts talking about the paper I was assigned, Large scale geometry of big mapping class groups by Katie Mann and Kasra Rafi. This is post “zero” in that series; I want to describe the project of large scale geometry of groups.

The Grothendieck Construction
July 6, 2022
I think that right after you learn the Yoneda lemma, the Grothendieck construction might be one of the most surprisingly useful techniques in basic category theory. The purpose of this post is to introduce the Grothendieck construction and illustrate a situation where it came up for me.

Adjoints
June 1, 2022
I realized I haven’t, to my knowledge, talked about adjoint functors yet! Let’s do that now, and introduce compactly generated topological spaces in the process.

Pursued By Stacks 7: Appendix
May 19, 2022
This post is, strictly speaking, a follow up to part three of my series on stacks, on the 2Yoneda lemma. The classical Yoneda lemma has built into it a naturality statement, while our 2Yoneda lemma proved essentially only the fact of the isomorphism. Let’s fix that.

Pursued By Stacks 6: Stacks And Groupoids
May 19, 2022
In writing this post, I started to see what’s kind of cool about stacks: you can treat them almost as if they were spaces in the sense that, as you’ll see, objects of a stack \(\mathsf{D}\) over a space \(Y\) are the same thing as maps of stacks \(Y \to \mathsf{D}\), just like the situation of \(\underline{X}\) for a space \(X\). The difference is that \(\mathsf{D}\) may have many isomorphisms stacked over \(Y \to \mathsf{D}\) while \(\underline{X}\) has only the identity of \(Y\).

Pursued By Stacks 5: Topological Stacks
May 19, 2022
The purpose of this post is to single out the topological (or geometric or Artin) stacks.

Pursued By Stacks 4: Fiber Products
May 18, 2022
Okay, we’re closing in on the definition of a geometric stack over \(\mathbf{Top}\). Actually, the nLab tells me such stacks are called topological, so I’ll try and say that. But! Before we get there, we need the construction of the 2fiber product of categories fibered in groupoids over \(\mathbf{Top}\). The purpose of this post is to discuss this construction, which I think is adorable. I cannot shake the feeling that I’ve talked through the 2fiber product of categories before, but I have no idea where or why I might have done this except possibly the last time I tried to learn about stacks.

Pursued By Stacks 3: We Need A Lemma
May 17, 2022
In the previous two posts we met stacks for the first time. We’d like to know what a geometric or Artin stack over \(\mathbf{Top}\) is, since these are the stacks that correspond to, e.g. graphs of groups. To get there, I’m told, Yoneda lemma (sorry sorry).

Pursued By Stacks 2: Descent
May 16, 2022
A stack, so I’m told, is a category fibered in groupoids satisfying some extra conditions. The stacks we’re interested in are geometric stacks, so this won’t be the last post on stacks, but with any luck we’ll see the definition in this post.

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.

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.

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.

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.

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 nonsingular Morse function on Outer Space and show that for a wellchosen 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 key tool to that end is the star graph, and the purpose of this post is to describe the star graph (for conjugacy classes) in a virtually free group and describe its usefulness.

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 \(T' \to T\). The set of \(G\)equivariant homeomorphism classes of trees in the same deformation space as \(T\) all of whose edges are surviving is partially ordered under the operation of forest collapse, and this partial order contains minimal elements, which are reduced trees. To study this poset, or its geometric realization \(L_T(G)\), it would be convenient to understand all trees collapsing onto a given reduced tree. This is captured in the idea of an ideal edge, which I’d like to describe in this post. The idea of an ideal edge goes back to Culler and Vogtmann’s original paper on Outer Space; the treatment here is adapted from a paper of Krstić and Vogtmann.

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. In this post I want to discuss the notion of a shelter, which Guirardel and Levitt (and also Clay) use to construct a deformation retraction of \(\mathscr{D}\) (or its simplicial spine) onto a smaller, often finitedimensional, space.

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.

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.” I explain Git just enough to get you started on a simple project, and hopefully enough so that you can start Googling things you’re unfamiliar with. I’m not an expert, and I expect my inexpertise to show.

Boundaries Of Free Splittings
July 25, 2021
The other day on Twitter I asked whether it might be the case that a nonproper 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 \(\mathbb{N}^\mathbb{N}\). In this post I’ll talk through why this is the case. There is also a notion of a Bowditch boundary for a tree with vertices of infinite valence. I’d like to show that the Bowditch boundary of such a tree is a Cantor set.

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 quasiisometry 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 quasiisometry in this post, which might make it useful for future reference. The reader already familiar with Gromov hyperbolicity and quasiisometries 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.

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 minicourse I gave.

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.

Pullbacks And Intersections
June 17, 2021
Sometimes a pullback square is an intersection! Here’s a short post about something cute that came up in my research this week.

Nielsen Realization
June 12, 2021
I’m giving a talk in Nearly Carbon Neutral Geometric Topology about Nielsen realization for infinitetype 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.

A Mathematical Postcard
June 9, 2021
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.

Notes On Orbifolds
March 14, 2021
The purpose of this post is to share some talk notes I made on orbifolds and graphs of groups as étale groupoids.

The Train Track Algorithm For Graphs Of Groups
December 21, 2020
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. At the end of the post I talk through using the train track algorithm to compute an example coming from a pseudoAnosov 5braid.

The Axiom Of Choice Is False For Manifolds
August 2, 2020
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.”

A Sheafy Adjunction
July 26, 2020
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.

Two Approaches To The Cotangent Space
July 10, 2020
There is a “sheaftheoretic” definition of something like a cotangent space at a point for a locallyringed space. In the case of the sheaf of differentiable functions on a smooth manifold, the two definitions—sheaftheoretic and the usual one in differential geometry—agree. What’s more, modulo the appropriate background technology in both subjects, the proof is beautiful and simple—I feel like shouting it from the rooftops at the moment, so this blog post will try and give a little exposition. I’ll try to move briskly without assuming total familiarity with either side of things.

Orbifolds Day One
July 5, 2020
I wrote some notes introducing orbifolds from Haefliger’s perspective in Metric Spaces of Nonpositive Curvature. There is some surprisingly heavy machinery that goes into it. In this blog post I want to talk a little about it at a high level.

Category Theory Day One
February 16, 2020
My officemate asked me to give a “second lecture” in category theory in Eminar, the Tufts graduate student seminar. I gave a first lecture last semester, but I thought it might be nice to have something to point people who missed it to, in order to stick to my assignment and begin at the Yoneda lemma. So, here is yet another rendition of the first hour of a category theory class.

Choo Choo
September 6, 2019
Today I posted my first paper, Train Tracks, Orbigraphs and CAT(0) Freebycyclic Groups to the ArXiv! Here is a description of the paper and the idea behind the main application.

A Digression
July 13, 2019
I’m not sure what this post set out to be. It ended up a digression on set theory vs. category theory.

The Category Of Graphs
June 29, 2019
You know the sensation where a familiar word becomes suddenly completely strange for no apparent reason? Like suddenly the scales fall away from your eyes and you see how utterly weird it is that a concept has a particular name? I’ve been having that feeling with the word graph as I use it in math.

Hello World
June 22, 2019
Earlier this week, Marc Weidenbaum of disquiet.com wrote a paean to the blog that I found really inspiring. “Selfpublishing,” he writes, “is the heart of the healthy internet.”