We revisit the geometry of involutions in groups of finite Morley rank. The focus is on specific configurations where, as in ${PGL}_2(\mathbb{K})$, the group has a subgroup whose conjugates generically cover the group and intersect trivially. Our main result is the subtle yet strong statement that in such configurations the conjugates of the subgroup may not cover all strongly real elements. As an application, we unify and generalise numerous results, both old and recent, which have exploited a similar method; though in fact we prove much more. We also conjecture that this path leads to a new identification theorem for ${PGL}_2(\mathbb{K})$, possibly beyond the finite Morley rank context.

中文翻译：

---

具有ti-subgroup的排名组中对合的几何

我们重新审视了有限Morley等级组中对合的几何形状。重点是特定的配置，例如${聚乳酸}_2（\mathbb{ķ}）$，该组具有一个子组，其共轭词通常覆盖该组，并且不容易相交。我们的主要结果是一个微妙而又强有力的陈述，即在这种构型中，亚组的共轭物可能无法覆盖所有强实元素。作为应用程序，我们统一并归纳了利用类似方法的许多旧的和最新的结果。尽管实际上我们证明了更多。我们还推测，这条路径导致了一个新的识别定理${聚乳酸}_2（\mathbb{ķ}）$，可能超出了有限的Morley等级范围。