Uncountably many $2$-spherical groups of Kac-Moody type of rank $3$ over $\mathbb{F}_2$

TL;DR

The paper proves that Weyl-invariant commutator blueprints of type $(4,4,4)$ over $\mathbb{F}_2$ are faithful and that integrability is equivalent to Weyl-invariance. This framework enables the construction of uncountably many non-isomorphic $RGD$-systems of type $(4,4,4)$ over $\mathbb{F}_2$, yielding $2$-spherical Kac-Moody groups over a finite field that are not finitely presented and showing that the local-to-global principle for $2$-spherical twin buildings fails. The results build a bridge between combinatorial blueprint data and geometric group-theoretic objects, expanding the landscape of explicit Kac-Moody-type groups with exotic finiteness and rigidity properties. Consequently, the work has implications for the theory of twin buildings, automorphism groups, and topologically simple completions, underscoring the limits of classical finite-presentability and extension theorems in this setting.

Abstract

In this paper we show that Weyl-invariant commutator blueprints of type $(4, 4, 4)$ are faithful. As a consequence we answer a question of Tits from the late $1980$s about twin buildings. Moreover, we obtain the first example of a $2$-spherical Kac-Moody group over a finite field which is not finitely presented.

Figures (13)

• Figure 1: Illustration of the group $V_R$
• Figure 2: Illustration of the group $O_R$
• Figure 3: Illustration of the group $H_R$
• Figure 4: Illustration of the group $J_{R, t}$
• Figure 5: Illustration of the group $G_R$

Theorems & Definitions (129)

• Theorem
• Theorem
• Theorem 1
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• Corollary 4
• Remark 1
• Theorem 5
• Remark 2
• Theorem 6