Irreducible subshifts associated with $\tilde A_2$ buildings

TL;DR

This paper studies a specific subshift of finite type derived from a $ ilde A_2$ building with a group acting transitively, proving its irreducibility using combinatorial properties of finite projective planes.

Contribution

It introduces a new subshift associated with $ ilde A_2$ buildings and proves its irreducibility, linking geometric group actions with symbolic dynamics.

Findings

The subshift of finite type is irreducible.

The proof relies on combinatorial properties of finite projective planes.

The work connects geometric group theory with symbolic dynamics.

Abstract

Let $\Gamma$ be a group of type rotating automorphisms of a building $\cB$ of type $\widetilde A_2$, and suppose that $\Gamma$ acts freely and transitively on the vertex set of $\cB$. The apartments of $\cB$ are tiled by triangles, labelled according to $\Gamma$-orbits. Associated with these tilings there is a natural subshift of finite type, which is shown to be irreducible. The key element in the proof is a combinatorial result about finite projective planes.