## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, J.-P. Bourguignon, M. Kotani, Y. Maeda and N. Tose, eds. (Tokyo: Mathematical Society of Japan, 2009), 321 - 334

### 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.

#### Article information

**Dates**

Received: 2 August 2007

Revised: 3 April 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543447919

**Digital Object Identifier**

doi:10.2969/aspm/05510321

**Mathematical Reviews number (MathSciNet)**

MR2463508

**Zentralblatt MATH identifier**

1183.19003

#### Citation

Pichot, Mikaël. The space of triangle buildings. Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, 321--334, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05510321. https://projecteuclid.org/euclid.aspm/1543447919