The space of triangle buildings

Abstract

I report on recent work of Sylvain Barré and myself on the space of triangle buildings.

From a set-theoretic point of view the space of triangle buildings is the family of all triangle buildings (also called Bruhat–Tits buildings of type $\tilde{A}_2$) considered up to isomorphism. This is a continuum. We shall see that it provides new tools and a general framework for studying triangle buildings, which connects notably to foliation and lamination theory, quasi-periodicity of metric spaces, and noncommutative geometry.

This text is a general presentation of the subject and explains some of these connections. Several open problems are mentioned. The last sections set up the basis for an approach via $K$-theory.