Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the groups are either: finite, or finite index, virtually free. This result applies, in particular, to systolic groups. prove similar strong restrictions extensions for other classes asymptotically The proof relies studying homotopy types at infinity question. also non-uniform lattices complexes (and more general complexes) not presentable and acting properly such act complexes. In Appendix we present topological two-dimensional quasi-Helly property