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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.