Let G be a group acting on a tree. I will discuss necessary conditions for G to have a finitely generated infinite normal subgroup of infinite index. When the edge stabilisers are virtually cyclic this naturally leads to considering (virtual) fibering of G. I will give an “if and only if” criterion for (virtual) fibering in the special case of amalgamated free products over virtually cyclic subgroups. The talk will be based on joint work with Jon Merladet.

The quasi-isometry group QI(X) of a metric space X is a natural group of automorphisms of the space that preserve its large-scale structure. The quasi-isometry groups of most familiar spaces are usually enormous and quite wild. Spaces X for which QI(X) is understood tend to exhibit a sort of rigidity phenomenon: every quasi-isometry of such spaces is close to an isometry. We exploit this phenomenon to address the question of which abstract groups arise as the quasi-isometry groups of metric spaces. This talk is based on joint work with Paula Heim and Joe MacManus.

Since the foundational works of Diaconis, pointwise character bounds of the form $\chi(\sigma) \le \chi(1)^\alpha$ have guided the study of growth in finite simple groups. However, this classical machinery hits an algebraic bottleneck when confronted with non-class functions and unstructured subsets.

In this talk, we bypass this barrier by replacing classical representation theory with discrete analysis. By decomposing functions as $f = \sum f_\rho$ and bounding the $L_2$ norm $\|f_\rho\|_2 \le \chi_\rho(1)^\alpha$ for each representation $\rho$, we develop a robust theory of Fourier anti-concentration. We will demonstrate how this resolves the Liebeck–Nikolov–Shalev (LNS) conjecture—proving a group can be expressed optimally as the product of conjugates of an arbitrary subset $A$—and discuss how applying Boolean function analysis tools like hypercontractivity pushes this philosophy even further.