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Flats and the flat torus theorem in systolic spaces. (English) Zbl 1228.20033
Summary: We prove the Systolic Flat Torus Theorem, which completes the list of basic properties that are simultaneously true for systolic geometry and CAT(0) geometry. We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries.

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M07 Topological methods in group theory
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