ZETAGROUPDYN_Zeta functions of groups and dynamical systems
Final Report Abstract
Under the project the Principal Investigator and a PhD student carried out fundamental mathematical research in the subject of Group Theory and its applications in the theory of Dynamical Systems. Group Theory is the study of symmetries, in an abstract algebraic sense. Elements g of a group G can act simultaneously and in a consistent manner as ‘symmetries’ on other mathematical objects. The object acted upon can be a model of a physical object of interest, such as a crystal, and often the object carries an intrinsic geometric or combinatorial structure that is to be preserved by the symmetries involved. In one strand of the project the Principal Investigator and his Co-Investigator in Poland studied analytic functions that encode geometric-combinatorial data coming from certain dynamical systems. The dynamical systems can be thought of as geometric objects equipped with a pair of self-maps, and the task is to understand synchronisation points, that is points whose orbits intersect under simultaneous iteration of the two maps involved. In another strand of the project the PhD student funded by the project and a fellow PhD student, also guided by the Principal Investigator, invented a completely new construction in the theory of groups acting on infinite regular rooted trees. The construction associates to any group G of automorphisms on an infinite regular rooted tree a new group Bas(G). This explains and interprets in a very general way the existence of a curious group, the ‘classical Basilica group’, that was discovered as an isolated example twenty years earlier and since then inspired a lot of research activity.
Publications
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On Denert's Statistic. The Electronic Journal of Combinatorics, 29(3).
Carnevale, Angela & Tielker, Elena
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Weighted Ehrhart Series and a Type-B Analogue of a Formula of MacMahon. Discrete & Computational Geometry.
Tielker, Elena
