  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 SimpHAtic groups are either: finite, or of
finite index, or virtually free. This result applies, in particular, to normal
subgroups of systolic groups. We prove similar strong restrictions on group
extensions for other classes of asymptotically aspherical groups. The proof
relies on studying homotopy types at infinity of groups in question. We also
show that nonuniform lattices in SimpHAtic complexes (and in more general
complexes) are not finitely presentable and that finitely presented groups
acting properly on such complexes act geometrically on SimpHAtic complexes.
