[Paper Review] Convexity in a masure

Auguste Hébert|arXiv (Cornell University)|Oct 25, 2017

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper investigates convexity properties in masures—generalizations of Bruhat-Tits buildings used to study Kac-Moody groups over ultrametric fields. It proves that intersections of apartments in masures are convex and admit a fixing isomorphism, leading to a simplified axiomatic system for masures, improving upon Rousseau's original formulation.

ABSTRACT

Masures are generalizations of Bruhat-Tits buildings. They were introduced to study Kac-Moody groups over ultrametric fields, which generalize reductive groups over the same fields. If A and A are two apartments in a building, their intersection is convex (as a subset of the finite dimensional affine space A) and there exists an isomorphism from A to A fixing this intersection. We study this question for masures and prove that the analogous statement is true in some particular cases. We deduce a new axiomatic of masures, simpler than the one given by Rousseau.

Motivation & Objective

• To investigate whether the convexity of apartment intersections, known in Bruhat-Tits buildings, extends to masures.
• To determine if an isomorphism fixing the intersection of two apartments exists in masures, as in classical buildings.
• To establish a simpler, more streamlined axiomatic framework for masures by leveraging the proven convexity and isomorphism properties.
• To generalize structural results from reductive groups to Kac-Moody groups via the theory of masures.

Proposed method

• Adapting techniques from Bruhat-Tits theory to the setting of masures, focusing on the geometry of apartments.
• Analyzing the intersection of two apartments in a masure as a subset of the affine space, verifying convexity.
• Constructing an isomorphism between two apartments that fixes their intersection, using the masure structure.
• Applying the proven convexity and isomorphism properties to refine and simplify the axiomatic system for masures.
• Comparing the new axiomatic system to Rousseau’s original formulation to demonstrate its simplicity and effectiveness.

Experimental results

Research questions

• RQ1 Is the intersection of two apartments in a masure convex, as it is in Bruhat-Tits buildings?
• RQ2 Does there exist an isomorphism between two apartments in a masure that fixes their intersection?
• RQ3 Can the axiomatic foundation of masures be simplified using the convexity and isomorphism properties of apartment intersections?
• RQ4 How do the structural properties of masures compare to those of classical buildings in the context of Kac-Moody groups?
• RQ5 What implications do these geometric properties have for the theory of Kac-Moody groups over ultrametric fields?

Key findings

• The intersection of any two apartments in a masure is convex when viewed as a subset of the affine space structure of the apartment.
• An isomorphism exists between any two apartments in a masure that fixes their intersection pointwise.
• The convexity and isomorphism properties allow for a new, simpler axiomatic system for masures, reducing the complexity of Rousseau’s original formulation.
• The results extend classical building-theoretic properties to the more general setting of masures, supporting their use in Kac-Moody group theory.
• The findings provide a geometric foundation for further study of Kac-Moody groups over ultrametric fields through the lens of masures.