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CAT(0) spaces and Gromov's condition

A powerful tool to generalize the concept of curvature to metric spaces is the comparison of triangles in them to those in model spaces. This notion of curvature, known as CAT($\kappa$) geometry, can be applied to different metric spaces. In this thesis, I concentrate on cube complexes on the one hand and give a simplified proof of Gromov's link condition, which relates the CAT(0) property to a combinatorical one. On the other hand, I transfer the CAT(0) geometry concepts to discrete settings and investigate its relation to defects in quivers. In this way, I show the existence of connections between the lack of positive vertex defect in quivers, chart defects and the CAT(0) property. In particular, a square lattice with defects fulfills the discrete CAT(0) inequality if and only if it does not harbour any positive vertex defects. Furthermore, I prove that the presence of chart defects is equivalent to the chart complex violating the CAT(0) property. Through these chart complexes, I find a link from quiver geometry to CAT(0) cube complexes. This connection can be used for further research through the application of the tools of CAT(0) cube complexes and their links to discrete setting.

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Year: 2022

Author : Antonia Seifert

Prof. Dr. Beatrice Pozzetti