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 non-uniform 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. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

中文翻译：

---

SimpHAtic 群的正规子群

如果一个群在几何上作用于一个简单连接的单纯遗传非球面 (SimpHAtic) 复合体，则它是 SimpHAtic。我们证明了 SimpHAtic 群的有限呈现正规子群是： 有限的，或 有限索引，或 几乎免费。 这一结果尤其适用于收缩组的正常亚组。我们证明了对其他类渐近非球面群的群扩展的类似强限制。证明依赖于研究无穷多组的同伦类型。我们还表明 SimpHAtic 复合体（以及更一般的复合体）中的非均匀晶格不是有限可表示的，并且有限表示的组正确作用于此类复合体在几何上作用于 SimpHAtic 复合体。在附录中，我们介绍了收缩复合物的拓扑二维准 Helly 特性。