Non-positive curvature and cubical surfaces

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The main goal of this project is to exhibit and to analyze high genus surfaces that appear embedded (or immersed) in higher-dimensional cubical manifolds. For this we build on techniques such as combinatorial holonomy concepts and branched coverings that were developed in the first funding period of Project JZ, "Combinatorial Holonomy". Additionally, our methods will use discrete concepts of combinatorial curvature in the sense of Alexandrov and Gromov (see [19]) in an essential way.In order to tackle known open problems about cubical (and other polyhedral) surfaces we want to access a wider class of interesting candidates (in particular, high curvature/high genus surfaces, and surfaces with extremal f-vector). For this, we first study and classify strongly regular combinatorial cubical surfaces (embedded or immersed) in certain higher-dimensional cubical manifolds. Then we can apply techniques developed in Project Z (3.7) to decide if the surfaces (to be) found can also be embedded into Euclidean space.