Abstracts
Naomi Andrew
Free-by-cyclic groups, actions on trees, and automorphisms
Free-by-cyclic groups depend only on an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and connect some to properties of the defining automorphism. I'll then discuss some ideas and techniques we can use to understand their automorphisms, by finding useful actions on trees, as well as related questions involving centralisers of elements of Out(F_n). (This is joint work with Armando Martino.)
Jonas Deré
Strongly scale-invariant virtually polycyclic groups
A finitely generated group is called strongly scale-invariant if there exists an injective endomorphism on the group with image of finite index and such that the intersection of the images of the iterations is trivial. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this talk, we will discuss this question in the special case of virtually polycyclic groups, by using the algebraic hull of these groups.
Sam Hughes
Irreducible lattices fibring over the circle
Let $n\geq 2$ and let $\Lambda$ be a lattice in a product of simple non-compact Lie groups with finite centre. An application of the Margulis Normal Subgroup Theorem implies that if $H^1(\Lambda)$ is non-zero, then $\Gamma$ is reducible. In the more general $\mathrm{CAT}(0)$ setting there are many irreducible lattices with non-vanishing first cohomology. In this case we can deploy the BNSR invariants and investigate how far these cohomology classes are from a fibration of finite type CW complexes. In this talk we will combine the groups of Leary and Minasyan with the technology of Bestvina and Brady to construct the first examples of irreducible lattices which fibre over the circle.
Ian Leary
Commensurating HNN-extensions of free abelian groups
Subgroups of the orthogonal group that preserve a lattice are finite, since they must permute the finitely many lattice vectors contained in any ball around the origin. However, there are elements of the orthogonal group of infinite order that commensurate lattices, and HNN-extensions constructed using these elements are CAT(0) groups with some unexpected properties: they are not biautomatic and give a negative answer to a question of Dani Wise. (Joint work with Ashot Minasyan.)
Yuri Santos Rego
Higher dimensional soluble groups and their twisted conjugacy classes
The group-theoretic property R∞, popularized in the last three decades, has remarkable applications in many areas, e.g. Lie theory, topology, and cohomology. Having R∞ means that all automorphisms of the given group have infinitely many twisted conjugacy classes. While it has been intensively investigated which nilpotent or polycyclic groups have property R∞, the case of soluble (non-nilpotent) groups remains rather mysterious. In this talk we will give a brief overview on twisted conjugation. Then, we shall consider soluble matrix groups and show that, in higher dimensions, many such groups do in fact exhibit R∞, heavily contrasting with soluble fundamental groups of low-dimensional manifolds. Based on joint work with Karel Dekimpe and Paula Macedo Lins de Araujo.