Hierarchically hyperbolic groups (HHGs) and spaces are recently introduced generalizations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG.
We consider some recent results on the weak commutativity construction suggested by Said Sidki in 1980. By definition given a group G and an isomorphic copy H of it we can consider a new group X(G) that is a quotient of the free product of G and H by the normal closure of the the commutators [g,h] where h is the image of g in H and g runs through the elements of G. We will consider some recent results about the structure of X(G) due to Bridson-Kochloukova, Kochloukova-Sidki, Lima-Oliveira.
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterization of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter.
Free groups, and free products of finite groups, are the easiest non-abelian infinite groups to think about. Yet the automorphism groups of such groups still present significant mysteries. We discuss a programme of research concerning automorphisms of easily understood infinite groups.
The question of whether a subgroup, given by generators, has finite (and then which) index is a natural question in group theory. Unfortunately, for natural groups such as SLn(ℤ) and Sp2n(ℤ), this question cannot have a general algorithmic solution. Nevertheless it is often possible to determine this information in many cases using a computer. I will describe some approaches to this problem and illustrate these in examples.
This is joint work with Alla Detinko (Hull) and Dane Flannery (Galway).
The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that the same does not hold for finitely presented groups with Dehn function as small as cubic. In particular, for every m≥2 and sufficiently large odd integer n, there exists an embedding of the free Burnside group B(m,n) into a finitely presented group with cubic Dehn function.
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