Triangle buildings and actions of type $III_{1/q^2}$

TL;DR

This paper investigates group actions on triangle buildings and their boundaries, analyzing associated von Neumann algebras, and identifies conditions under which these actions are hyperfinite of specific type.

Contribution

It introduces new results on the classification of boundary actions of triangle buildings and their von Neumann algebra types, especially for buildings of order q ≥ 3.

Findings

Boundary actions are hyperfinite of type III_{1/q^2} for buildings of order q ≥ 3

Provides classification of von Neumann algebras arising from these actions

Establishes connections between geometric structures and operator algebra types

Abstract

We study certain group actions on triangle buildings and their boundaries and some von Neumann algebras which can be constructed from them. In particular, for buildings of order $q\geq 3$ certain natural actions on the boundary are hyperfinite of type $\tqs$.