CAREER: Kleinian, Arithmetic and Mapping Class Groups

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In this project the PI will study three particular classes of groups, namely Kleinian groups, mapping class groups and lattices in higher rank Lie groups. In spite of being quite different, these groups show remarkable similarities. This may be explained by the rigidity properties of associated geometric objects such as locally symmetric spaces, moduli space,... The goal of this project is to study how the rigidity of these associated geometric objects is reflected in the algebraic properties of the groups in question. More concretely, the PI will study the relations between the number of generators of a Kleinian group and geometric observables of the associated hyperbolic 3-manifold, such as the volume or the spectrum of the Laplacian. In order to do so, it will be necessary to develop some so-far unexplored aspects of the deformation theory of Kleinian groups. The PI will also study to what extent the Margulis superrigidity theorem holds for homomorphisms between mapping class groups. This problem will be approached via combinatorial methods and also via the study of harmonic maps between moduli spaces. Tools from dynamics will also be applied to study the rigidity of certain actions of mapping class groups. Finally, the PI will study the existence of minimal spines for locally symmetric spaces and their geometric, homological and algebraic properties. In this case, the main tools are going to be a combination of classical arguments in algebraic topology and in the theory of algebraic groups.A group is an algebraic object of fundamental interest. Perhaps the most interesting groups appear as symmetries of some geometric structure, such as a crystal, a physical model, an object in space, etc... For instance, Kleinian groups play an important role in the study of fractal objects. Arithmetic groups are groups of matrices and hence have applications in all branches of mathematics. Finally, the mapping class group, the group of symmetries of a surface, plays not only a central role in mathematics, but is also important in for instance theoretical physics. All three classes of groups are known to have useful rigidity properties but it is unknown how this rigidity is effectively interlocked with the algebraic properties of the groups. It is an integral part of this proposal to obtain effective and, at least in principle, computable rigidity results. Besides advancing the state of knowledge in an important area of mathematics, educating graduate students to do independent research is one of the main goals of this project. This will be greatly facilitated by the broad spectrum of tools and methods that the PI intends to apply. This breadth will also facilitate the disemination of the obtained results among the mathematical community.