- Flats and flat torus theorem in systolic spaces
Abstract: We prove that that minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for such surfaces which implies minimality. We also prove that minimal surfaces are stable under small modification of their boundaries. These results are used to establish the Flat Torus Theorem for systolic complexes.
- Systolic groups with isolated flats
Abstract: We study possible configurations of flats in an arbitrary systolic complex. We apply the results to prove that a systolic complex has the Isolated Flats Property if and only if it does not contain isometrically embedded triplanes. We also show that systolic groups with the Isolated Flats Property are relatively hyperbolic with respect to their maximal abelian subgroups of rank at least 2 and that they satisfy the Relative Fellow Traveler Property.
- Isometries of systolic spaces
Abstract: We prove that any isometry of a systolic complex is either elliptic or hyperbolic. This dichotomy has several consequences on groups acting on systolic complexes.