[Paper Review] Compact Polygons

Linus Kramer|arXiv (Cornell University)|Apr 5, 2001

Homotopy and Cohomology in Algebraic Topology20 citations

TL;DR

This paper investigates compact polygons—compact topological Tits buildings of rank two—establishing that their Coxeter diagrams are always crystallographic, restricting existence to n=3, 4, or 6. It classifies transitive compact polygons as Moufang, linking them to real Lie groups of rank 2.

ABSTRACT

We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive group action, showing that such a polygon is Moufang and thus related to a real Lie group of rank 2.

Motivation & Objective

• To establish the topological properties of compact polygons, defined as compact topological Tits buildings of rank two.
• To determine the conditions under which compact n-gons can exist, particularly focusing on their Coxeter diagrams.
• To classify compact polygons admitting a transitive group action and understand their structural implications.
• To investigate the connection between such polygons and real Lie groups of rank 2, especially in the Moufang case.

Proposed method

• Analyzing the topological and combinatorial structure of compact polygons as rank-two buildings.
• Applying the theory of Coxeter groups and their diagrams to constrain possible polygon types.
• Using the classification of Moufang polygons to identify those compact polygons supporting transitive group actions.
• Leveraging the theory of topological buildings and their associated root systems to relate compact polygons to Lie groups.
• Establishing that crystallographic Coxeter diagrams are necessary for compact polygon existence.
• Employing group-theoretic techniques to show that transitive action implies Moufang structure.

Experimental results

Research questions

• RQ1 For which values of n do compact n-gons exist as topological Tits buildings of rank two?
• RQ2 What constraints do the Coxeter diagrams of compact polygons impose on their structure?
• RQ3 Which compact polygons admit a transitive group action, and what structural properties do they possess?
• RQ4 How are compact polygons with transitive group actions related to real Lie groups of rank 2?
• RQ5 Under what conditions is a compact polygon Moufang, and what does this imply for its automorphism group?

Key findings

• Compact n-gons exist only for n = 3, 4, or 6, as these are the only values for which the associated Coxeter diagrams are crystallographic.
• The Coxeter diagram of any compact polygon is necessarily crystallographic, ruling out non-crystallographic types.
• Any compact polygon admitting a transitive group action is Moufang, indicating a rich algebraic structure.
• Such Moufang compact polygons are classified by their connection to real Lie groups of rank 2, such as PGL(3,R) or PSL(2,R).
• The existence of a transitive group action forces the polygon to be Moufang, linking topology and algebraic group theory.
• The classification of transitive compact polygons reduces to the known classification of Moufang polygons over real closed fields.