Minimal displacement set for weakly systolic complexes

In this paper, we investigate the structure of the minimal displacement set in weakly systolic complexes. We show that such set is systolic and that it embeds isometrically into the complex. As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycenter of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).