This is a post mostly about AI, I guess, and to complain about a bad definition which somehow managed to stay relevant in group theory despite having very few known consequences for twenty-five years or so.

Dynamical Systems, Surjunctivity

Suppose $X$ is a space and $\rho \colon G \times X \to X$ is an action of $G$ on $X$ by homeomorphisms. A continuous map $\Phi \colon X \to X$ is $G$-equivariant if for each $g \in G$ and $x \in X$, we have $\Phi(\rho(g,x)) = \rho(g,\Phi(x))$.

For example, $G$ might be the integers under addition, and $X = \mathbb{R}$, where the action is $\rho(k,x) = x + k$. In this setting, you can check that the maps $\Phi_1, \Phi_2\colon \mathbb{R} \to \mathbb{R}$ defined as

$\Phi_1(x) = \pi x \qquad \Phi_2(x) = x + \pi$

have the property that $\Phi_1$ is not equivariant, but $\Phi_2$ is.

Suppose $S$ is some finite set (typically called an “alphabet”). Let $S^G$ denote all the functions $\omega \colon G \to S$. This is also the product of copies of $S$ indexed by $G$. If we give $S$ the discrete topology and $S^G$ the product topology, then by Tychonoff’s theorem, since $S$ is compact (it’s finite and discrete), so too is the (typically infinite) space $S^G$. Any group $G$ always admits an action on this space by the rule $\rho(g,\omega)(h) = \omega(g h)$.

Walter Gottschalk called a group $G$ surjunctive if every equivariant continuous map $S^G \to S^G$ which is injective is automatically surjective.

Surjunctivity is sort of like finiteness: just by counting, you can see that any function from a finite set to itself which is injective is also surjective. On the other hand, it is much more mysterious: To my knowledge, there is still no known example of a non-surjunctive group.

There are actually lots of properties in mathematics which similarly evade our grasp. To name a couple off the top of my head: We do not know whether every finitely presented group with one end is semistable at infinity. We do not know whether every word hyperbolic group is residually finite. We do not know whether every finitely generated purely pseudo-Anosov subgroup of the mapping class group is also convex-cocompact. We do not know whether every Artin group is torsion free. We do not know whether every mapping class group (of a surface group of finite type) is linear.

Sofic groups

A finitely generated group is sofic (forgive me for handwaving) if it has a Cayley graph with respect to a finite generating set which can be approximated to any desired specificity by finite graphs.

Explicitly, such a Cayley graph $\Gamma$ has edges labeled by the generating set. We seek for each $r > 0$ and $\epsilon > 0$ a finite graph $(V,E)$ with edges similarly labeled such that there is a subset $V_0 \subset V$ of size at least $(1 - \epsilon)|V|$ such that the $r$-neighborhood of each vertex $v_0 \in V_0$ is isomorphic (respecting labels) to the $r$-neighborhood of any vertex in $\Gamma$.

This is, I must say, a bad definition. The reason it is the way that it is is best explained by introducing ultraproducts, which are gorgeous things that I can’t waste your time with today.

The reason for the definition (perhaps), is twofold: on the one hand, one can prove that sofic groups are surjunctive. It was Gromov who did this first, in 1999. The second reason, which is quite pretty, but sort of difficult to explain quickly, is that two important but obviously limited families of groups are easily seen to be surjunctive: the amenable groups and the residually finite groups.

A group $G$ is amenable if every continuous action on a compact space $X$ admits an invariant probability measure. The “volume” (Lebesgue) measure on a unit ball is an example of a probability measure. A math-pilled reader may be aware of the Banach-Tarski paradox which doubles the volume of a 3-dimensional ball. Amenability was introduced by John von Neumann to name the phenomenon allowing (or disallowing) similar paradoxes to occur in various settings.

A group $G$ is residually finite if whenever $g \in G$ is nontrivial, there is a homomorphism $\varphi\colon G \to H$ with $H$ finite so that $\varphi(g)$ is still nontrivial.

These two properties are ubiquitous and useful in group theory, but it is not difficult to come up with non-examples. If it were the case that every group is sofic, it would follow that every group is surjunctive. Weiss, naming the term “sofic”, writes

It is not likely that all groups are sofic — but I don’t know of any definite example of a non sofic group. A concrete case that I haven’t been able to resolve is the universal Burnside group on a finite set of generators.

How to feel about non-sofic groups

Recently, OpenAI announced an example of a non-sofic group. OpenAI calls this a “central open question in group theory”. Respectfully, and I’m so sorry Alex Lubotsky, I do have to disagree. Many, if not most papers about sofic groups cannot be bothered to really define them. Similar problems exist for open questions in K-theory with group theoretic interest, so this is not a problem unique to sofic groups, but it’s hard to defend a claim that such a question is central if its definition remains largely inaccessible.

I was heartened to read the Leiden declaration on AI and Mathematics recently. The declaration is clear-headed, and does a good job of steering clear of both hype and Luddism. OpenAI doesn’t really know what to do about this declaration, since the paper is very clearly not written by somebody who can understand the model’s output. This is a shame. They write

We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.

I do have to ask, though: what ideas?

In terms of the existence of a non-sofic group, like most research-level results I’ve seen so far from LLMs, this one resolves the easier and less sexy (forgive me) direction of a question. A proof that all finitely generated groups are sofic would necessarily be rather difficult, since the stated hypotheses give you nearly nothing to work with. OpenAI’s disproof just needs one non-sofic group.

I skimmed the non-sofic groups chapter of the OpenAI paper fairly quickly. The claimed group is apparently a group of units of a certain algebra. The algebra is simple to write down, but the model neglects to give a presentation of the claimed non-sofic group in a way that caught my eye. The method of (dis)-proof seemed more or less straightforward, but I wasn’t able to follow it mostly because I had other things to do with my day.

I think the deep idea here is this: we love our conjectures too much to give them the Riemann hypothesis treatment. The Riemann hypothesis, asking about the pattern of zeros of the zeta function, has been tested extensively for more or less every reasonably computable value (at least, such is my non-expert understanding). Reading the AI disproofs we’ve seen so far, it appears to me that many well-known conjectures have not seriously been subject to the same sort of treatment.

I know a little bit about how this goes! Here’s a little story.

Over the lockdown era, one of my projects was about ends of the groups $\mathrm{Out}(A_1 * \cdots * A_n * F_k)$, where the $A_i$ are all finite and $F_k$ is free of finite rank $k$. I was following methods of Karen Vogtmann, with input from a paper of Collins and Zieschang with the delightful title Rescuing the Whitehead method for free products.

Vogtmann’s ideas got me pretty far: relying on her work I was able to reduce the problem to three cases. One case showed up in her work as well, but two were new, and it was these two that really had me stumped for a while. Since I wanted my proof to apply to all the relevant groups above, I kept wanting to choose some extra bit of information that was not guaranteed to exist.

Finally, one weekend in the fall after spending yet another summer more or less stumped, I decided to sit down and spend the weekend attempting to prove the opposite of my desired statement. To my great surprise, I found a beautiful little pattern waiting for me that showed that the very smallest case I was worried about did indeed not work out. That example, together with some supporting material, is my paper When is the Outer Space of a free product CAT(0)?. After a bit more hemming and hawing, I did manage to get the rest of the cases of my initial theorem to work out; that project is One-endedness of outer automorphism groups of free products.

More recently, having not learned my lesson, an LLM was able to set me straight on a joint project where we became enamored of a possibility which would be pretty and difficult to prove.

Living without a moat

I’ll close by offering some thoughts about AI in general. In software companies, apparently, there is this concept of a “moat”. The moat is what keeps your lunch from being stolen by another company willing to undercut you on cost. Sometimes your moat is a facet of your software, other times maybe it is a costly form of regulatory approval you’ve undergone.

Many people share their worries with me about AI as it impacts their work as mathematicians, musicians or programmers. A common thread I hear in all of these fears is a sort of identification of their worth as a person (at least, in an economic sense) being tied to skills that they are suddenly feeling like no longer provide them with much of a moat. Here are some things to ponder in that regard.

• You never had a moat. I have not met anyone who is inarguably the single best person in their field. Even among exceptionally talented people, many of whom I haven’t met, many are either neck-and-neck with their “competition”, or have a niche where they’re better which doesn’t extend to the whole field. It’s really easy to tie your self-work to the worth of the work that you do: I do it too, all the time. It hurts every time for me too, so I’m trying not to.
• You do have the work, and your team. If we accept the above as true, in our extremely connected era, perhaps the real reason that pure disinterested economics has not pushed all of us out of our jobs is that the job was always about more than the skillset. For example, my work as a university professor is not just about research or giving lectures on calculus. How I am doing at my job is also about connections I am able to build with and for my students, and support I’m able to give my colleagues.
• The reason for the work is not volume of production. Many jobs which have aspects that are automatable are still alive and kicking. For example, although most clothing on earth is produced in bulk by machines and invisibilized workers typically far from where I live in the US, many, if not most of the people working in fashion design know how to sew, and this skill serves them some purpose even if their work does wind up mass-produced.
• First time? If we’ve discovered that some of the purpose of the work that people did without LLMs has benefits not directly correlated to the output of the work (e.g. coding to figure out a problem space, even if you throw the code away before writing a final solution), we may have found some aspects of the work which are more like art or literature or craft. Many people continue to learn and build their skills in fields like this long after the market has “killed” it as a profession, and some do continue to have it be their main source of income. Once AI could produce coherent videos, why did the film industry not implode? I’m sure I don’t have a full answer, but it does appear that our culture values the skills involved in making a movie enough to pay a number of people to do that work full time.
• “Theorem credits” was always the wrong metric. Similarly, number of users on a given product, for example, is well-known to not correlate with the quality of the product. We have a whole new word about that, “enshittification”. If you haven’t read Bill Thurston’s On Proof and Progress in Mathematics recently, you might find it comforting to read again.

Now, if you’ll forgive me, I have much more interesting stuff, which was even written by humans, to go read.