An obstruction relating locally finite polygons to translation quadrangles

TL;DR

This paper introduces a category of embeddings of generalized polygons that links two longstanding open problems in Incidence Geometry, potentially providing new insights into their solutions.

Contribution

It proposes a novel categorical framework that connects the existence of locally finite polygons with properties of translation quadrangles.

Findings

The category controls the existence of locally finite generalized polygons.

It relates the endomorphism rings of translation quadrangles to algebraic structures.

Provides a new approach to longstanding open problems in Incidence Geometry.

Abstract

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.