Short Research Presentations

Thursday

Heiko Dietrich

Martin Kassabov

Gabi Nebe

Tobias Rossman

Christopher Voll

Christian d’Elbée

Emmanuel Rauzy

Frank Wagner

Marina Conte

Marco Amelio

Jeroen Schillewaert

Heiko Dietrich (Monash University): The group isomorphism problem is easy for almost all group orders

We discuss why the title is true for finite groups in the Cayley table model.

Joint work with James Wilson (FOCS 2022).

(Pending permission to publish the recording)

Martin Kassabov (Cornell University): Presentations of Simple Groups of Ree type

We show what the finite simple groups of Ree type have presentations with a bounded number of generators and relations independent of the size of the field. A starting point of the argument is the Steinberg presentation which comes from viewing 2G2(Fq) as a rank 1 group. The key step is reducing the number of relations from q3 to a bounded number. This uses algebraic geometry over rings with extra endomorphism, and showing that certain varieties have points defined over the finite field Fq.

Joint work with Alexander Hulpke, Akos Seress, and James Wilson.

Gabi Nebe (RWTH Aachen): On the natural involution in group algebras of finite groups

Given a semisimple rational algebra A with an involution ι, what are the conditions for A to be the group algebra A ∼= QG for some finite group G where ι : g 7 → g−1 is the natural involution?

We computed invariants of the restriction of the natural involution to direct summands of QG for the ATLAS groups up to the Harada–Norton group and for many finite groups of Lie type (joint with Richard Parker, Thomas Breuer, David Schlang and Linda Hoyer).

There are certain surprising facts for adjoint involutions of an invariant quadratic form: For some sporadic groups the quadratic extension of the character field determined by the discriminant of the invariant quadratic forms is not Galois over the rationals.

We never observed square classes having an odd dyadic valuation, motivating Richard Parker to conjecture that these discriminants are always odd. This conjecture is a theorem for solvable groups and for some infinite series of finite groups of Lie type; however we do not have a structural explanation for this phenomenon.

(Pending permission to publish the recording)

Tobias Rossman (University of Galway): Enumerating conjugacy classes of graphical groups

Each choice of a (finite, simple, undirected) graph and commutative ring gives rise to an associated graphical group. I will report on recent developments surrounding the symbolic enumeration of conjugacy classes of graphical groups over natural infinite families of finite rings.

(Pending permission to publish the recording)

Christopher Voll (Bielefeld University): Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

In this talk, I would like you to meet Hall-Littlewood-Schubert series, a new class of multivariate generating functions. Their definition features semistandard Young tableaux and polynomials resembling the classical Hall-Littlewood polynomials. Their intrinsic beauty notwithstanding, Hall-Littlewood-Schubert series have many applications to counting problems in algebra, geometry, and number theory. In my talk the spotlight will be on applications to affine Schubert series. These may be seen as an integral analogue of the Poincaré polynomials enumerating the rational points over finite fields of classical Schubert varieties. The latter parametrize subspaces of a given vector space by the intersection dimensions with a fixed flag of reference. I will explain things from scratch, assuming no familiarity with the advanced technical vocabulary used in this abstract.

Joint with Joshua Maglione.

Christian d’Elbée (Universidad del Pais Vasco): Lie methods for ω-categorical Engel groups

A structure M (group, Lie algebra, associative algebra, etc.) is ω-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is ω-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M ) on the cartesian power M^n, for all natural number n. In 1981, Wilson conjectured that any ω-categorical locally nilpotent p-group is nilpotent. If true, a quite satisfactory decomposition of every ω-categorical group would follow. This conjecture is very much open more than 40 years later.

The analogue statement for Lie algebras (every locally nilpotent ω-categorical Lie algebra is nilpotent) is also open. Both statements can be reformulated as: any ω-categorical Engel group/Lie algebra is nilpotent. As such, those questions are connected to Burnside-type problems and the work of Higman, Kostrikin, Zelmanov, Vaughan-Lee, Traustason, etc. For instance, using a classical result of Zelmanov, the conjecture for Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over Fp is nilpotent for all but finitely many p. There is a similar statement for groups. The situation for small values of the pair (n, p) is highly characteristic-dependent, for instance, 4-Engel Lie algebras over a field of characteristic p are nilpotent except if p = 2, 3 or p = 5. I recently proved that ω-categorical n-Engel Lie algebras over a field of characteristic p are nilpotent for (n, p) = (3, 5) and (n, p) = (4, 3). Lie methods can then be used in order to deduce that 5-Engel 5-groups are nilpotent, and I will briefly explain the main ingredients for those results.

Emmanuel Rauzy (Munich): Effective residual finiteness and a question of Higman

Being able to algorithmically list the finite quotients of a finitely generated group provides a way to use algebraic properties (like residual finiteness, subgroup separability, or profinite rigidity), to obtain solution to decision problems (here, the word problem, the membership problem or the isomorphism problems). Recent results of Nies related to computability of profinite groups provide new reasons to explore such questions. I will present results from my PhD about computability of finite quotients of finitely generated group, and a new result obtained with Arman Darbinyan: there is a finitely generated residually finite group with solvable word problem that does not embed in the group of computable permutations of the natural numbers.

(Pending permission to publish the recording)

Frank Wagner (Université Claude Bernard Lyon 1): Skew Braces of small Morley Rank

A skew brace is a structure which carries two group laws + and ∗, related by a left distributive law a(b + c) = ab − a + ac. They were introduced in an attempt to analyze solutions to the set-theoretic Yang–Baxter equation from theoretical physics, but have increasingly been studied for their algebraic properties. I shall present a classification of skew braces of Morley rank at most three.

This is joint work with my student Moreno Invitti, as well as Maria Ferrara (Caserta) and Marco Trombetti (Naples).

Marina Conte (Bielefeld University): Definability of the dimension of R-analytic groups

Building from recent work of Nies, Segal and Tent, in joint work with Benjamin Klopsch we showed that the dimension of p-adic analytic pro-p groups is definable by one sentence in the language of groups. In joint work in progress with Andoni Zozaya we consider more general R-analytic groups, where R is any pro-p domain, and investigate the definability of their dimension. In this talk I will introduce the groups under consideration and give an overview of this problem.

(Pending permission to publish the recording)

Marco Amelio (Münster University): Non-split sharply 2-transitive groups of odd characteristic

Until recently, the existence of non-split sharply 2-transitive groups (i.e., sharply 2-transitive groups without a normal abelian subgroup) was an open problem. The first examples of such groups were exhibited by Rips, Segev and Tent in 2017 and by Rips and Tent in 2019. It is possible to associate to every sharply 2-transitive group a characteristic that is either 0 or a positive prime number. The first of these examples were in characteristic 2, while the others were in characteristic 0, leaving the problem open for odd characteristics. In this talk, I will outline an adaptation of the construction in characteristic 0 to build examples of non-split sharply 2-transitive groups in odd characteristic using methods of geometric small cancellation.

This is joint work with Simon André and Katrin Tent.

(Pending permission to publish the recording)

Jeroen Schillewaert (University of Auckland): Discreteness and freeness of 2-generator subgroups of SL_2(K), where K is a local field

I will give a brief overview of some recent results obtained jointly with Matthew Conder in the non-archimedean case, by my student Ari Markowitz over the reals, and jointly with Alex Elzenaar and Gaven Martin over the complex numbers.

(Pending permission to publish the recording)