Epimorphisms of generalized polygons A: The planes, quadrangles and hexagons

TL;DR

This paper classifies epimorphisms from finite thick generalized polygons to thin ones for m=3, 4, 6, and explores the distinct nature of infinite cases, introducing new structural theories.

Contribution

It provides a classification of epimorphisms for finite cases and develops new theories for infinite generalized polygons, highlighting their differences.

Findings

Finite cases classified for m=3, 4, 6

Infinite cases exhibit fundamentally different behavior

Introduction of locally finitely generated and chained generalized polygons

Abstract

Inspired by a theorem by Skornjakov-Hughes-Pasini [9, 7, 8] and a problem which turned up in our recent paper [13], we start a study of epimorphisms with source a thick generalized m-gon and target a thin generalized m-gon. In this first part of the series, we classify the cases m = 3, 4 and 6 when the polygons are finite. Then we show that the infinite case is very different, and construct examples which strongly deviate from the finite case. A number of general structure theorems are also obtained. We introduce the theory of locally finitely generated generalized polygons and locally finitely chained generalized polygons along the way.