Infinite groups of nonpositive curvature

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The research program described in the proposal concerns groups and geometric spaces that can be thought of as intricate mixtures of (portions of) flat planes and negatively curved pieces, assembled together along prescribed local singularities. The project is to explore the properties of these groups and spaces, classify them, provide new constructions, and study invariants associated with them. The proposal contains a list of open problems and new ideas to achieve these goals, including for example on spaces of "intermediate rank", random spaces, random groups, or on Bruhat-Tits buildings and their automorphism groups. The unifying theme is the theory of infinite groups, scrutinized from a geometric standpoint, following Gromov, and especially involving geometry of nonpositive curvature. This often naturally brings in adjacent domains of mathematics, like dynamical systems or operator algebras, and several of these connections are investigated too. For example, many open problems on dynamical systems arising from measure preserving actions of these groups on probability spaces are described in the proposal, especially concerning their orbit structures, their invariants (cost, isomperimetric constant, sofic dimension), or their rigidity properties. The theory of operator algebras will play a important role. We will be studying invariants of infinite groups and dynamical systems, and many of them (e.g. L2 Betti numbers or operator K-theory) are operator algebraic in nature. Other closely related areas include graph theory and computer science. The latter has both a theoretic component in algorithmic and a software programming component. In particular, a very tangible output will be a computer program on infinite groups of nonpositive curvature that will help classifying them and understanding their properties. The proposal includes a range of open problems with various degrees of difficulty, from classical problems that are exciting and provide guidance to (presumably) more contained open problems that can be approached by students at various level. Educational activities form an essential aspect of this proposal. Students working on these questions will be able to contribute to progress in the field and open their own research perspectives.