Automorphisms of Lie incidence geometries with spectral gaps

TL;DR

This work characterizes uniclass automorphisms of thick irreducible spherical buildings of simply laced type via a spectral-gap property on the associated long root subgroup geometries, proving that an automorphism is uniclass iff it is a $\{1,2'\}$-kangaroo. The authors provide a comprehensive case analysis across classical types $\mathsf{A}_n$ and $\mathsf{D}_n$ and exceptional types $\mathsf{E}_6$, $\mathsf{E}_7$, $\mathsf{E}_8$, showing how fixed Weyl substructures and opposition data govern uniclass behavior, and they demonstrate failures in non-simply laced cases via counterexamples. The paper also extends the kangaroo framework to other Lie incidence geometries (polar, metasymplectic) and derives additional classifications for $\{2,2'\}$-kangaroos, providing a suite of applications and strengthening the link between spectral gaps and fixed-point geometry in buildings. Overall, the results yield a clear, architecture-level criterion for uniclass automorphisms in the simply laced setting and illuminate the geometric underpinnings of automorphism spectra in Lie incidence geometries.

Abstract

An automorphism of a building is called uniclass if the Weyl distance between any chamber and its image lies in a unique (twisted) conjugacy class of the Coxeter group. In a previous paper we characterised uniclass automorphisms of spherical buildings in terms of their fixed structure. In the present paper we restrict to the simply laced case and characterise uniclass automorphisms in terms of a spectral gap property. More precisely, we show that an automorphism of a thick irreducible spherical building of simply laced type is uniclass if and only if no point of the long root subgroup geometry is mapped to distance $1$ or codistance $1$.

Theorems & Definitions (52)

• Theorem A
• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Proposition 1.4
• Theorem 1.5: NPV
• Theorem 2.1
• proof
• Proposition 2.2: NPVV
• Theorem 2.3