Below you can find descriptions of 3 upcoming preprints, which are all part of one project, and are also of independent interest. The descriptions are aimed at a wide mathematical audience, sometimes at the expense of completeness and accuracy.
Below you can find descriptions of 3 upcoming preprints, which are all part of one project, and are also of independent interest. The descriptions are aimed at a wide mathematical audience, sometimes at the expense of completeness and accuracy.
(upcoming preprint 2025)
This paper is dedicated to the proof of several results concerning random walks on algebraic groups. Here I will only describe the applications to expander graphs.
Background: It is challenging to construct highly connected finite graphs with many vertices and few edges. By "few edges" we mean that the graph is K-regular. That is, each of the N vertices has only K neighbours (say K=4). By "highly connected" we mean that the graph is a C-expander for some C>0 not very close to zero. This means that for every partition of the vertex set V = S ∪ T, there are at least C min{|S|,|T|} edges with one endpoint in S and the other in T.
As a (counter) example, consider a cycle graph on 2N vertices. Then K=2 and C=2/N, so C is very small when N is large. We are looking for the opposite of that! We want sequence of larger and larger graphs X1, X2, X3,... all K-regular for the same K, and all C-expanders for the same C>0. Such a sequence is called a family of expander graphs. There are no such families for K=2, but there they do exist for every fixed K≥3 (a highly non-trivial fact!).
There is a vast body of research of expander graphs and their applications. The first step, in the late 60s, was to show that expanders exist without providing a method to construct them. Explicit constructions were first found in the 70s, with applications to computer science (e.g. coding theory and hardness of approximation) starting to appear in the 90s. Then, came the question of whether specific graphs arising in mathematics are expanders. A positive answer gives insight in the context where the graphs arise (for example, the fact that some particular graphs are expanders may prove that a particular algorithm works well, as in [Kaluba–Kielak–Nowak]).
This paper: The present paper proves a result of the latter type: In order to achieve our goal in our paper Random Character varieties (see below), we needed to know that some particular graphs were expanders. This is achieved in the present paper, which is of independent interest as it solves an instance of the Lubotzky–Weiss uniformity problem. As with most explicit constructions of expanders, our graphs of interest are Cayley graphs: For a group G and a subset S of G, the Cayley graph Cay(G, S) is the graph with G as its vertex set (i.e. one vertex per element of the group G), and the neighbours of vertex g are precisely the vertices of the form s⋅g and s-1⋅g for s∊S. This graph is K-regular for K=2|S|.
Specifically, we study Cayley graphs Cay(G, S) where G is SL3(Fp) or a similar group. This is the group of 3x3 matrices with determinant 1 and entries in the finite field Fp, p prime. In fact, the result works for much more general groups (i.e. perfect linear algebraic groups - see the paper for details), and in particular you can take any fixed d≥2 instead of d=3, but let's keep considering SL3(Fp) for the sake of concreteness. The very first explicit construction of expanders (1973) showed that if you carefully choose a subset Sp of the group SL3(Fp) for every prime number p, then the sequence (Cay(SL3(Fp), Sp))p prime is a family of expanders. Our result is simple to state (not simple to prove..): You don't have to be careful! Any choice of the subsets Sp works, as long as it makes the the graphs Cay(SL3(Fp), Sp) connected. There is a caveat: Our proof works for almost every prime number p, not quite all prime numbers p. With this caveat in mind, our result is a dichotomy: Every Cayley graph Cay(SL3(Fp), Sp) of the group SL3(Fp) is either disconnected or highly connected (depending on the choice of Sp). There is no middle ground! It remains an open problem whether the result is true for all prime numbers p (the conjecture is that it does), but our application in our paper Random Character Varieties does not require this stronger statement: "almost every prime" is good enough.
(upcoming preprint 2025)
Group theory is the mathematical framework for the study of symmetry. For example, the collection of symmetries of any given geometric object forms a group. The study of random groups deals mainly with the question: What does a typical group look like? The answer depends on the choice of a probabilistic model for random groups. We study a particular aspect of a random group Γ, namely, its representation variety Hom(Γ, G), where G is a fixed complex semisimple Lie group (e.g. G=SL2(C)) and Hom(Γ, G) is the space of group homomorphisms from Γ to G. The representation variety is an algebraic variety (i.e. the geometric shape cut from space by a system of simultaneous polynomial equations), and we study its geometric properties: dimension, irreducible components and the action of the absolute Galois group. More precisely, we study the principal part Z(Γ, G) of the representation variety, consisting of all homomorphisms whose image is Zariski-dense in G (i.e. not trapped in an algebraic hypersurface inside G). In the context of this research, the appropriate probabilistic model for random groups turns out to be the few-relators model: Fix integers k≥2 and r≥1, and choose r random words w1,...,wr of length L over the alphabet s1 ±,...,sk ±. The resulting random group is Γ = <s1,...,sk | w1,...,wr>, the group generated by the formal letters s1,...,sk subject to the relations w1=1...,wr.=1. We give a formula for the dimension of the variety Z(Γ,G), determine whether it is irreducible as a function of k and r, and bound the nummber of irreducible components when it is not irreducible. All of these results hold in high probability, i.e. with probability tending to 1 as the relator-length L tends to infinity (in fact, the probability of exceptions decays exponentially as L grows). Some of our results depend on the Generalized Riemann Hypothesis. Our paper opens the door to several future directions in the study of random groups. For example, it follows from our work that when r=k-1 (say, 2 generators and 1 relator), all of the Zariski-dense homomorphic images of Γ in G are isomorphic to each other, and there is at least one such image. This image is a quotient of Γ, and it would be fascinating to study it. A similar statement holds more generally when r≤k-1, while Z(Γ,G) is empty (with high probability) when r≥k. Our proof relies on our results in the two other papers described on this page.
(upcoming preprint 2025)
The results in this paper are important ingredients in our proofs in the two papers described above.
Background: An affine algebraic variety X is the set of simultaneous solutions in Fm to a system of polynomial equations in several indeterminants (F an algebraically closed field). We consider two algebraically closed fields: the field C of complex numbers, and the algebraic closure Fpal of the finite field Fp (for any given prime number p). If all of the coefficients in the polynomial equations are integers then they are also in C, and thus they give rise to an algebraic subvariety X of Cm . But we may also reduce the coefficients of the polynomial equations modulo a prime number p and obtain an algebraic subvariety Xp of (Fpal)m. We say that Xp is the reduction modulo p of X.
Now, every algebraic variety decomposes (uniquely) as a union of irreducible subvarieties, analogously to how whole numbers decompose as products of prime numbers. Starting from X, the paper will compare to things one can do: we may either reduce X modulo p to obtain Xp, and then decompose Xp as the union of its irreducible components, or we could decompose X as the union of its irreducible components and then reduce each of them modulo p (well, the irreducible components of X are not necessarily defined by polynomial equations with integer coefficients, but we can take a prime ideal P lying over p in the ring of integers of a large enough number field, and reduce the irreducible components of X modulo P).
This paper: We have described above two processes, each resulting in a collection of irreducible subvarieties of (Fpal)m: Either first reduce modulo p and then decompose, or first decompose and then reduce modulo p. It turns out that the two collections are the same for all but finitely many bad prime numbers p. One result of our paper is an effective upper bound on the number of bad primes: The bound is polynomial the the maximal degree among the defining polynomials of X, and logarithmic in the maximal absolute value of their coefficients (here the ambient dimension m is considered constant). The quality of this bound is crucial for our applications. We also bound the discriminants of the fields of definition of the irreducible components of X, and other related quantities.