Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over $\Z$

For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^λ(R)$ corresponding to an integrable representation $V^λ$ with dominant integral weight $λ$. When $R=\Z$, we prove that this map extends to a group homomorphism $ρ_{λ,\Z}: G(\Z) \to G^λ(\Z).$ We prove that the kernel $K^λ$ of the map $ρ_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $φ:G(\Z)\to G(\C)$ is injective, then $K^λ\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.

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