  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 $G^{\lambda}(R)$ corresponding to an integrable representation
$V^{\lambda}$ with dominant integral weight $\lambda$. When $R=\Z$, we prove
that this map extends to a group homomorphism $\rho_{\lambda,\Z}: G(\Z) \to
G^{\lambda}(\Z).$ We prove that the kernel $K^{\lambda}$ of the map
$\rho_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group
homomorphism $\varphi:G(\Z)\to G(\C)$ is injective, then $K^{\lambda}\leq
H(\Z)\cong(\Z/2\Z)^{rank(G)}$.
