Normal Subgroup Theorem for groups acting on $\tilde A_2$-buildings

TL;DR

The paper proves a Normal Subgroup Theorem for cocompact lattices in buildings of type $\tilde{A}_2$, showing that every nontrivial normal subgroup has finite index. It develops a comprehensive measure-theoretic framework for spaces of embeddings via prouniform measures, and constructs several flows (Cartan, harmonic, singular, and detecting) to analyze boundary dynamics. Central to the argument is a Factor Theorem for Γ-equivariant measurable factors of the boundary, derived from contracting actions on the boundary and the projectivity group, which yields amenability of quotients and rigidity results. The work also establishes strong consequences: either the lattice is virtually simple or residually finite, and it lays groundwork for broader rigidity phenomena and potential generalizations to other 2-dimensional buildings. The combination of incidence-geometry, ergodic theory, and measured embedding spaces provides a geometric path to Margulis-type rigidity beyond Bruhat-Tits settings.

Abstract

Let $Γ$ be a group acting on with finite stabilizers and finite fundamental domain on a building of type $\tilde A_2$. We prove that any non-trivial normal subgroup of $Γ$ is of finite index in $Γ$.

Figures (3)

• Figure 1: The convex hull of $[0,v)$ and $\lambda_{m,n}$
• Figure 2: The model space for the detecting flow on a tree
• Figure 3: The model space for the detecting flow

Theorems & Definitions (142)

• Theorem 1
• Corollary 2
• proof
• Theorem 3
• Definition 1.1
• Example 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• proof