The automorphisms of a group do not vary particularly transparently with the group: for instance, changing a single integer in the presentation of a Baumslag–Solitar group can turn a finite outer automorphism group into one which is not even finitely generated. However, in nice enough situations, one can extract information about the outer automorphism group from investigating the ways the original group acts on trees. The challenge becomes ensuring you are in such a situation, and interpreting the information you get.
I'll discuss these ideas and how they play out in work on the outer automorphism groups of (some) free-by-cyclic groups. (Joint with Armando Martino)
I will describe an explicit geometric construction of finitely presented groups ($F_\infty$, in fact) that contain infinite families of mapping class groups of compact manifolds (resp. automorphism groups of free groups and arithmetic groups). This is joint work with K.-U. Bux, J. Flechsig, N. Petrosyan and X. Wu.
Thanks to a celebrated decomposition theorem established by J. R. Stallings in the late 60s of the last century one knows that a finitely generated group $G$ with more than one end either decomposes as non-trivial free product with amalgamation over a finite subgroup or it decomposes as a non-trivial HNN-extension over a finite subgroup. If $G$ is a Coxeter group $(W,S)$ with more than one end, then $G$ is generated by involutions and thus every group homomorphism $\phi \colon W \to \mathbb Z$ is trivial. Hence HNN-decompositions cannot occur. As pointed out by Y. Cornulier in a Math-Overflow blog, the question which remains is whether a decomposition as $W \cong A \coprod_C B$ takes necessarily place in the category of Coxeter groups. We provide an affirmative answer to Y. Cornulier's question, based on results of M. W. Davis, and deduce some consequences on Coxeter groups of virtual cohomological dimension $\leq 1$ and groups acting Weyl–transitively on locally finite buildings.
The talk is based on a work in progress with Bianca Marchionna and Thomas Weigel.
Say we are given only the $R$-algebra structure of a group ring $RG$ of a finite group $G$ over a commutative ring $R$. Can we then find the isomorphism type of $G$ as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of $kG$ as a ring determine the isomorphism type of $G$ as a group, if $G$ is a $p$-group and $k$ a field of characteristic $p$?
Starting with an overview on the state of knowledge on general Isomorphism Problems and the modular one in particular, I will present a negative solution found rather recently, but also present positive structural results and several problems remaining open.
D. G. Higman observed that the centraliser algebra of a rank 3 permutation groups is spanned by the incidence matrix of a strongly regular graph on which the permutation group acts by automorphisms. Quite generally, incidence matrices of combinatorial structures invariant under a permutation group $G$ live in the centraliser algebra of the corresponding permutation representation. In this talk, we will consider instead monomial representations and their centraliser algebras, and discuss computational techniques for deciding whether such an algebra contains a complex Hadamard matrix. This is joint work with Ronan Egan, Heiko Dietrich and Santiago Barrera-Acevedo. Time permitting we will discuss connections to group cohomology and joint work with Dane Flannery, Assaf Goldberger and Giora Dula.
The relational complexity of a permutation group was introduced in model theory, and is a measure of the extent to which partial information about the action of a group element determines the existence of the element. A base of a permutation group $G$ is a sequence of points whose pointwise stabiliser is trivial. There are intimate connections between relational complexity and various types of bases. This talk will give a survey of recent activity in this rapidly-developing area.
I'll talk about two recent results on the enumeration of orbits of unipotent groups over finite fields. In both cases, graph-theoretic constructions play key roles. What sets the two projects apart is that in one case, the numbers of orbits turn out to be extremely tame (namely polynomial in the size of the field), whereas in the other case they end up being "approximately wild" in a geometric sense.
Coxeter groups are the abstract counterparts of classical (spherical, Euclidean, hyperbolic) reflection groups. By definition, any such group arises from a presentation attached to certain labeled graphs–the Coxeter-Dynkin diagram or the defining graph. Surprisingly, the natural question of which diagrams give rise to the same Coxeter group is still open. In this talk we shall explore the Coxeter galaxy–this is a space encoding Coxeter groups and their diagrams, up to isomorphism. We will look at some ways to navigate this galaxy and discuss which parts are well-understood so far. Based on joint work with Petra Schwer.
The concept of "special" presentations was introduced by Howie as a particular class of $C(3)-T(6)$ (small cancellation) presentations, specifically those whose relators have length $k = 3$ and whose star graph is the incidence graph of a finite projective plane. Edjvet and Vdovina generalized this to $(m, k)$-special presentations and we generalize it to $(m, k, \nu)$-special presentations. These are group presentations in which each relator has length $k$ and whose star graph has $\nu$ isomorphic components, each of which is the incidence graph of a generalized $m$-gon. Groups defined by $(m, k, \nu)$-special cyclic presentations act on Euclidean or hyperbolic buildings. A cyclic presentation of a group is a group presentation with an equal number of generators and relators that admits a cyclic symmetry. In this talk I will discuss work with Ihechukwu Chinyere in which we classify the $(m, k, \nu)$-special cyclic presentations and investigate SQ-universality of the groups they define.