On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow

TL;DR

This paper investigates when discrete groups acting on affine buildings are linear, establishing new results for exotic $ ilde{A}_2$-buildings and linking linearity to arithmeticity using ergodic theory tools.

Contribution

It proves that cocompact lattices in exotic $ ilde{A}_2$-buildings are non-linear and connects linearity of groups to arithmeticity via the singular Cartan flow.

Findings

Cocompact lattices in exotic $ ilde{A}_2$-buildings are non-linear.

Linearity implies arithmeticity for groups acting on Bruhat--Tits buildings.

Provides the first infinite family of lattices in exotic $ ilde{A}_2$-buildings of arbitrary large thickness.

Abstract

Let $X$ be a locally finite irreducible affine building of dimension $\geq 2$ and $\Gamma \leq \mathrm{Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $\Gamma$ linear? More generally, when does $\Gamma$ admit a finite-dimensional representation with infinite image over a commutative unital ring? If $X$ is the Bruhat--Tits building of a simple algebraic group over a local field and if $\Gamma$ is an arithmetic lattice, then $\Gamma$ is clearly linear. We prove that if $X$ is of type $\widetilde A_2$, then the converse holds. In particular, cocompact lattices in exotic $\widetilde A_2$-buildings are non-linear. As an application, we obtain the first infinite family of lattices in exotic $\widetilde A_2$-buildings of arbitrarily large thickness, providing also a partial answer to a question of W. Kantor from 1986. We also show that if $X$ is Bruhat--Tits of arbitrary type, then the linearity of $\Gamma$ implies that $\Gamma$ is virtually contained in the linear part of the automorphism group of $X$, in particular $\Gamma$ is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic $\Gamma$-space attached to the the building $X$, which we call the \emph{singular Cartan flow}.