将局部有限多边形与平移四边形联系起来的障碍
An obstruction relating locally finite polygons to translation quadrangles
摘要：
关联几何中最基本的公开问题之一是由Tits在20世纪60年代提出的，要求存在所谓的“局部有限广义多边形”。|即，具有“混合参数”（一个是有限的，另一个不是）的广义多边形。在一个更专门的背景下，另一个长期存在的问题（从20世纪90年代开始）是关于任何平移广义四边形的自同态环是否是除环（在有限情况下答案是已知的）。(The类似的问题，射影平面，其正解，“布鲁克-玻色结构”，是整个理论的基础。）在这个简短的说明中，我们介绍了一个类别，代表某些非常具体的嵌入广义多边形，这令人惊讶地控制这两个（显然完全无关）问题的解决方案。
英文摘要：
One of the most fundamental open problems in Incidence Geometry, posed by Tits in the 1960s, asks for the existence of so-called "locally finite generalized polygons" | that is, generalized polygons with "mixed parameters" (one being finite and the other not). In a more specialized context, another long-standing problem (from the 1990s) is as to whether the endomorphism ring of any translation generalized quadrangle is a skew field (the answer of which is known in the finite case). (The analogous problem for projective planes, and its positive solution, the "Bruck-Bose construction," lies at the very base of the whole theory of translation planes.) In this short note, we introduce a category, representing certain very specific embeddings of generalized polygons, which surprisingly controls the solution of both (apparently entirely unrelated) problems.
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