## Research

“Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root and it would be hard to find a better one on which to demonstrate the working of mathematical intellect.”
– Hermann Weyl

My research is mainly in group theory, and my recent work has mainly focussed on group generation, probabilistic group theory and permutation group theory. Two key themes of this work are the subgroup structure and conjugacy of (finite and infinite) simple groups, which are topics that have many applications across mathematics. However, my broader interests also include the representation theory of groups and geometric group theory.

For a list of my publications, see the Publications page.

### The 3⁄2-Generation Conjecture

Every finite simple group can be generated by just two elements. This theorem requires the Classification of Finite Simple Groups, and the fact that the finite simple groups of Lie type are 2-generated was proved by Steinberg (1962), by concretely exhibiting a generating pair for each group. In that paper, Steinberg wrote “It is possible that one of the generators … can be chosen as an arbitary element other than the identity” but noted that if true “would quite likely require methods much more detailed than those used here.” This inspires the following definition.

• Definition
A group is 32-generated if every nontrivial element is contained in a generating pair.

Guralnick and Kantor (2000), using probabilistic methods, proved that every finite simple group is indeed 32-generated. It is easy to see that a 32-generated group has the property that every proper quotient is cyclic, and Breuer, Guralnick and Kantor (2008) conjectured that this straightforward condition exactly characterises the finite 32-generated groups. In recent work with Burness and Guralnick (2021) we proved this conjecture.

• Theorem (Burness, Guralnick, H | Ann. of Math. | 2021)
A finite group is 32-generated if and only if every proper quotient is cyclic.

In fact, we proved a much stronger version of this conjecture. The spread of a group G, written s(G), is the greatest integer k such that for any k nontrivial elements x1, …, xk of G there exists an element y of G such that for each i we have 〈xi, y〉 = G. Therefore, G is 32-generated if and only if s(G) ≥1 . In 1975, Brenner and Wiegold, who introduced the notion of spread, proved that if a soluble group G satisfies s(G) ≥ 1, then, in fact, s(G) ≥ 3. They posed the problem of classifying the groups G such that s(G) = 1, suggesting there might be only finitely many. This problem has now eventually been solved.

• Theorem (Burness, Guralnick, H | Ann. of Math. | 2021)
A finite group G is satisfies s(G) ≥ 2 if and only if every proper quotient is cyclic. In particular, there are no finite groups G such that s(G) = 1.

These theorems are the culmination of an intense recent study of the generation properties of the almost simple groups of Lie type. Indeed, building on the work of Burness and Guest (2013) who proved them for almost simple linear groups, in two papers (2017, 2021) I proved these theorems for all almost simple classical groups. The joint paper with Burness and Guralnick (2021), features both a reduction of these theorems to the almost simple groups and a proof of the theorems for exceptional groups of Lie type (almost simple groups whose socles are alternating or sporadic were already known to satisfy the theorems).

The work on almost simple groups of Lie type involved a variety of methods, including the technique of Shintani descent from the theory of algebraic groups, a detailed analysis of the subgroup structure of these groups, and a probabilistic method via fixed point ratios, the final of these has been fruitful method for solving various problems across mathematics, from monodromy of Riemann surfaces to finite geometry.

You can watch a talk I gave at the Isaac Newton Institute, Cambridge on this and related topics in February 2020: video.

### Thompson's Group V

Thompson's group V naturally arises as a group of homeomorphisms of Cantor space, and alongside its siblings T and F, it has been the focus of much research, with connections across a broad spectrum of topics, such as dynamics and the word problem. This group was the first known example of a finitely presented infinite simple group and until 2000 all known examples of such groups were closely related to V, such as the Brin–Thompson groups mV and the Higman–Thompson groups Vn'.

Characterising the finite 32-generated groups (see above) and the other natural questions about the generation of finite simple groups that it raises, has generated much recent research interest and several fascinating results. Inspired by this, Donoven and I began thinking about the situation for infinite groups. The infinite cyclic group and the pathological Tarski monsters provide (in some sense trivial) examples of infinite 32-generated groups, and until recently these were the only known examples.

• Theorem (Donoven, H | Bull. London Math. Soc. | 2020)
Thompson's group V is 32-generated. Moreover, all of the Brin–Thompson groups mV and the Higman–Thompson groups Vn and Vn' are 32-generated.

### Uniform Domination and Bases

Let G be a nonabelian finite simple group. Then G is 2-generated and, moreover, Guralnick and Kantor proved that G is 32-generated, that is for all x ∈ G \ 1 there exists y ∈ G such that 〈x, y〉 = G. This raises the question: if you give me a nontrivial element x, how do I find a suitable element y? Guralnick and Kantor actually proved that y can always be chosen from a fixed conjugacy class. For example, if n > 13 is prime, then for all x ∈ An \ 1 there exists an n-cycle y in (1 2 … n)An such that 〈x, y〉 = An. However, (1 2 … n)An is a huge conjugacy class; do we need all of it? Burness and I proved that, in fact, you only need two of these elements! That is, there exists an element g such that for all x ∈ An \ 1 there exists y ∈ { (1 2 … n), (1 2 … n)g } such that 〈x, y〉 = An. Moreover, almost all choices of g give this result.

• Definition
The uniform domination number of G, written γu(G), is the least size of a set S of conjugate elements such that for all x ∈ G \ 1 there exists y ∈ S such that 〈x, y〉 = G.

• Theorem (Burness, H | Trans. Amer. Math. Soc. | 2019)
There are infinitely many finite simple groups with γu(G) = 2, for example, An for all prime n > 13, PSLn(q) for all odd n > 3, E8(q) and the Monster.

Indeed, many finite simple groups have this remarkably strong property, and in a follow up paper [Israel J. Math | 2020], Burness and I obtained a near classification of such groups, and several stronger probabilistic results. However, γu(G) is not bounded above by a constant for all simple groups G.

• Theorem (Burness, H | Trans. Amer. Math. Soc. | 2019)
If n ≥ 6 is even, then log2(n) < γu(An) < 2 log2(n).

The uniform domination number of a group is intimately connected to the base size of its primitive actions. More precisely, if G contains an element s that is contained in a unique maximal subgroup H of G, then the uniform domination number γu(G) is at most the minimal size of a base for G in the transitive action on G/H. Bases of permutation groups have been studied since the earliest days of group theory and they have also been the focus of recent major conjectures (e.g. Pyber's conjecture and Cameron's conjecture).

### Symmetry Breaking

You have a keyring with five keys that you cannot tell apart by eye. Every time you put down your keys and pick them up later, you lose track of which key was which. To distinguish the keys you decide to paint them different colours. However, you only have three colours at your disposal (including the natural colour of the keys, I guess). If you colour them around the ring as red, blue, blue, blue, yellow, then you can still tell each key apart (to distinguish the three blue keys note that one is next to a red key, one is next to a yellow key and one is between two blue keys). However, if you only had two colours, then I claim that you'd be stuck. No matter how you coloured them, I could sneakily rotate or reflect your keyring and you wouldn't be able tell, thus making you mix up your keys. In short, you need three colours to break all symmetries of the keyring and thus be able to distinguish the keys.

This captures the key idea of symmetry breaking: what is the most efficient way to modify a structure or system in order to remove all of its (nontrivial) symmetries? This idea plays an important role in many disciplines, including the construction of fast solution-searching algorithms (e.g. timetabling).

• Definition
The distinguishing number of a graph Γ, written D(Γ), is the least number of colours required to colour the vertices of Γ such that no nontrivial automorphism of Γ stabilises the colouring.

We have just seen that D(five-cycle) = 3 and you can check that D(n-cycle) = 2 if n is at least 6. This is a well studied idea in graph theory, and since it is really a question about groups, it is natural to make a more general definition.

• Definition
The distinguishing number of a permutation group G ≤ Sym(Ω), written D(G), is the smallest k for which there exists a partition of Ω with k parts such that subgroup of G stabilising each part of the partition is trivial.

Therefore, D(Γ) is nothing but D(Aut(Γ)).

Since D(G) = 1 if and only if G = 1, the most interesting case is where D(G) = 2, which is equivalent to G having a regular orbit on the power set of Ω. This suggests, that the distinguishing number is an interesting invariant of groups in its own right. It is. For example, replacing “partition” with “subset” (and “part” with “point”) in the definition yields the familiar definition of base size. Indeed, a study of the distinguishing number of quasiprimitive groups played an important role in the recent work on Pyber's conjecture on bases.

Cameron, Neumann and Saxl (1984) proved that if G ≤ Sym(Ω) is a primitive permutation group other than Sym(Ω) or Alt(Ω), then D(G) = 2 or G is one of 43 exceptions of degree at most 32 (when D(G) ≤ 4). Inspired by this, we proved the following:

• Theorem (Devillers, H, Morgan | Arch. Math. | 2019)
Let G ≤ Sym(Ω) be a semiprimitive group that is not primitive. Then D(G) = 2 or G is GL(2, 3) acting on the 8 nonzero vectors of GF(3)2 (in which case D(G) = 3).

Our result implies that several natural families of groups satisfy D(G) = 2 (i.e. have a regular orbit on the power set). For example, except for a handful small exceptions, D(G) = 2 for transitive actions of a simple group G except Alt(n) (acting naturally), D(GL(n, q)) = 2 for the action on nonzero vectors of GF(q)n, and D(Γ) = 2 for all connected noncomplete nonbipartite 2-arc-transitive graphs Γ.

You can watch a talk I gave at the BIRS Casa Matemática, Oaxaca on this topic in 2018: video|slides.

### Fractals and Multifractals

If you throw two darts at the unit interval how far apart do you expect them to be? It's a straightforward exercise to see that the answer is one third. However, the unit interval is quite dull. Let's throw darts at fractals! For example, what is the average distance between two points chosen uniformly at random from the middle-third Cantor set? By constructing recurrence relations based on the self-similarity of the middle-third Cantor set one can show that this average distance is two-fifths. More generally, in work joint with Allen, Edwards and Olsen (2016) we can found an exact formula for the average distance between points in a general self-similar Cantor set with respect to a general self-similar measure.

What more can we say about the distribution of distances between points in such sets with respect to such measures? We know the average distance, but what about higher moments of the distance function? Consider the middle-(1 ⁠–⁠ 2r) Cantor set C and the Bernoulli measure μ supported on C with probability vector (p, q).

• Theorem (Allen, Edwards, H, Olsen | Math. Z. | 2015)
Let Mn be the nth moment ∫C2 |x – y|n d(μ × μ)(x, y). Then log Mn / log n → –s as n → ∞, where s is the Renyi dimension log(p2 + q2) / log r. Moreover, nsMn = Π(n) + εn, where εn → 0 as n → ∞ and Π is a particular periodic function defined in terms of a zeta function.