## Oligomorphic groups: topology or geometry?

One perhaps unexpected result of the pandemic is that there is a huge volume of really interesting mathematics flying around the internet at the moment, courtesy of Zoom and other platforms.

This week I went to a talk by Joy Morris in which the view was expressed (not by her) that “Algebraic Graph Theory” means “Cayley graphs of finite groups”. The next day I heard a talk by Matt Zaremsky in which it was said, maybe not quite so forcefully, that “Geometric Group Theory” means “Cayley graphs of finitely generated infinite groups”.

I had gone along to this talk on embedding groups in simple groups because it looked interesting. I remembered some results along these lines from my Oxford days, when Graham Higman proved the Boone–Higman Theorem and talked about it in his advanced class. It was indeed a very interesting talk. Every finite group is embeddable in a finite simple group (this is Cayley’s Theorem with a very small twist); the game is to replace “finite” in this theorem by some other property. Philip Hall proved this for finitely generated groups. Matt talked about some work he has done with my colleague James Belk.

Let me describe briefly the groups they use. These are generalisations of Thompson’s group V, which is a certain group of homeomorphisms of Cantor space. You can think of Cantor space as the set of all infinite sequences of zeros and ones, with the product topology. It is a fractal, that is, it can be cut into two pieces each homeomorphic to the original space in a canonical way. (These are just the sequences starting with 0 and the sequences starting with 1.) Thompson’s group is defined as follows. Do this division m times, for some m (that is, bisect the whole space, then choose a piece and bisect it, and so on.) Then do it again starting from the top to get a possibly different set of m pieces. Now take the canonical isomorphisms between these pieces, and put them together; this gives an element of the group.

The first variation is that you can do this instead with s-dimensional Cantor space, simply the direct product of s copies of the original Cantor space. Now each bisection should be done to one coordinate. For s = 2, thinking of Cantor space as a square, you bisect it by a line parallel to one of the axes, and then repeat this with each piece. This gives the group known as sV.

In the previous paragraph, it seems as if s is finite, but it is allowed to be infinite here. I will write SV, where S is the index set.

For the second variation, we take a group G which acts faithfully on the set indexing the coordinates. Do the divisions as before, but now apply an arbitrary element of G to each piece before taking the canonoical maps between them. (In the 2-dimensional case, flip some of the pieces by interchanging horizontal and vertical.) This gives a simple group called SVG. It is clear that G embeds into SVG (take subdivisions with m = 1); if both G and SVG are finitely generated, then the embedding is a quasi-isometry (that is, it doesn’t distort too much the metric defined by the Cayley graph).

The next job is to investigate finiteness conditions of these groups. A group is said to be of type Fn if it acts “geometrically” on an (n−1)-connected CW-complex. It is of type F if it is of type Fn for all n. It turns out that type F1 means “finitely generated” while F2 means “finitely presented”. So can we find an action of a group G such that SVG has property Fn?

Now here came a big surprise for me. Their theorem says that, if the permutation group G on a countable set S is oligomorphic (diligent readers will recall that this means that G has only finitely many orbits on k-sets, for all natural numbers k), and the stabiliser of any finite set in G is of type Fn, then SVG is also of type Fn; if in addition G is not of type Fn+1, then neither is SVG. Thus if we can find suitable G we get a simple construction of simple groups with any finiteness condition Fn.

Now came the second surprise. I will explain later why this was surprising to me. An example of an oligomorphic group satisfying the requirements of this theorem is the Houghton group Hn. This group acts on the disjoint union of n copies of the natural numbers N; apart from a finite subset, the action on the ith copy is a translation by, say, ai (where, for consistency, these translation steps must sum to 0).

Why is the Houghton group oligomorphic? Well, any finite set can be permuted arbitrarily, so it contains the finitary symmetric group; it actually has the much stronger property of being highly transitive.

The first surprise was that here were my old friends, oligomorphic groups, in quite a different context from the one where I usually meet them. The second surprise went deeper. The kind of oligomorphic groups I am most friendly with are things like the automorphism group of the random graph. This group is simple, but it is uncountable, hence certainly not finitely generated; so it does not fall within the domain of geometric group theory at all!

The difference is that I think of oligomorphic groups as being groups of automorphisms of relational structures like the random graph. Taking the closure in the symmetric group (in the topology of pointwise convergence) does not change the orbits on k-sets; but if the group is closed and oligomorphic then it is certainly uncountable. So my “topologically nice” oligomorphic groups are completely different from the “geometrically nice” groups which are required by the theorem of Belk and Zaremsky.

Is there any common ground here?

The Houghton group Hn is commonly drawn as acting on the set of equally-spaced points on n rays radiating from the origin in the plane. Looked at “from a distance”, we see only the translations, which form a group Cm−1; whereas, close up, we see no structure at all, arbitrary permutations of a finite set, and a highly transitive group. So it is all in the point of view.

I would like to construct an oligomorphic group which is interesting at both scales. So let me try to formulate this as a precise question. Does there exist an oligomorphic permutation group on a finite set which has property Fn for some but not all finite n and is not highly transitive?

The remarks above hint that such a group should contain finitary permutations. Now Wielandt’s extension of a theorem of Jordan says that if a primitive permutation group on an infinite set contains finitary permutations then it contains the finitary alternating group (the set of all finitary even permutations), and so is highly transitive. So indications are negative!

However, all is not lost. Dugald Macpherson has constructed (alone or with co-authors) various interesting actions of finitely generated free groups. These could at least give examples where the theorem could construct simple groups of type F, which. Perhaps applying these methods one could get other examples.

As a final note, Matt Zaremsky observed that he was not sure how to pronounce “oligomorphic”: is the initial O short or long? It is certainly short in British English, but I don’t know about the US version. How do you pronounce “oligarch”, as in “friend of Vladimir Putin”?

The word “oligomorphic” is not in my Chambers dictionary, but I understand that it has another usage, in computer science. A computer virus is oligomorphic if it exists only in a few different forms; such viruses are more easily recognised than the polymorphic ones.

Which brings us back to Covid …

$\bar{o}$, ‘cept in Texas, where we pronounce “oil” as “all” …