The spectrum of infinite monster groups

ended

Groups are abstract algebraic structures that mathematically encode the notion of symmetry. Groups are ubiquitous in all areas of mathematics and have applications, for example, to theoretical physics and computer science. Group theory tries to understand particular classes as well as global properties of groups. The theory of infinite monster groups is a crucial part of the theory. It provides examples of infinite groups with exceptional geometric, analytic, and algebraic properties, thus clarifying boundaries of classes of groups and often resolving outstanding open questions. This project aims to develop further methods for constructing and studying infinite monster groups, providing new techniques for producing such groups as well as a deeper understanding of the spectrum of phenomena encountered in the existing theory. The monsters within the scope of this project arise from methods of geometric group theory, which studies finitely generated infinite groups through their actions on geometric structures. The planned work will benefit greatly from the excellent synergies between the ER's geometric viewpoint and Prof. Thom's more analytic viewpoint on infinite groups, a combination which has produced outstanding results in the past. The methods and results developed in this project will lead to significant advances in outstanding open questions. Particular topics addressed by the project are, for example: Diximier's problem on unitarizability of groups (open since 1950), the Kaplansky zero-divisor conjecture (1956), the Baum-Connes conjecture (1982), and the open questions of residual finiteness of hyperbolic groups and of quasi-isometry invariance of acylindrical hyperbolicity. Furthermore, the project will lead to new models of random groups, in particular infinitely presented ones and periodic ones.