Harmonic cochains and K-theory for $\widetilde A_2$ groups

TL;DR

This paper investigates the K-theory of boundary C*-algebras associated with torsion-free A_2 groups acting on A_2 buildings, revealing a precise relationship between K_0 groups and second cohomology.

Contribution

It establishes an explicit isomorphism between the realified K_0 group of the boundary algebra and the second cohomology dimension for A_2 groups acting on buildings.

Findings

K_0(k A_A_2 groups) d7 b R \u2261 d7 b R^{2eta_2}

Dimension of K_0 tensor d7 b R equals twice the second cohomology dimension

Provides a link between K-theory and group cohomology for A_2 groups

Abstract

If $\Gamma$ is a torsion free $\widetilde A_2$ group acting on an $\widetilde A_2$ building $\Delta$, and $\fk A_{\Gamma}$ is the associated boundary $C^*$-algebra, it is proved that $K_0(\fk A_\Gamma)\otimes \bb R \cong \bb R^{2\beta_2}$, where $\beta_2=\dim_\bb R H^2(\Gamma, \bb R)$.