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Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finitefield $\mathbb{F}_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the$q+1$--homogeneous tree. Let $\Gamma$ be a nonuniform lattice in $G$. When$\Gamma=P_i^-$, $i=1,2$, the standard parabolic subgroup for the negative$BN$--pair, the quotient graph $P^-_i\backslash X$ is the positive half of thefundamental apartment of $X$, a semi-infinite ray. We define Eisenstein serieson $P_1^-\backslash X$. We prove convergence of Eisenstein series in a halfspace. This uses Iwasawa decomposition of the Haar measure on $G$. A crucialtool is a description of the vertices of $X$ in terms of Iwasawa cells, whichwe give. We prove meromorphic continuation of Eisenstein series using theSelberg--Bernstein continuation principle. This requires an analog of integraloperators on the Tits building and the classical truncation operator forEisenstein series, which we construct.

Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields, with an appendix by Paul Garrett