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Motivated by seminal paper of Kozlov et al.(1975) we consider in this paper abranching process with a geometric offspring distribution parametrized byrandom success probability $A$ and immigration equals $1$ in each generation.In contrast to above mentioned article, we assume that environment isheavy-tailed, that is $\log A^{-1} (1-A)$ is regularly varying with a parameter$\alpha>1$, that is that ${\bf P} \Big( \log A^{-1} (1-A) > x \Big) =x^{-\alpha} L(x)$ for a slowly varying function $L$. We will prove thatalthough the offspring distribution is light-tailed, the environment itself canproduce extremely heavy tails of distribution of the population at $n$-thgeneration which gets even heavier with $n$ increasing. Precisely, in thiswork, we prove that asymptotic tail ${\bf P}(Z_l \ge m)$ of $l$-th population$Z_l$ is of order $ \Big(\log^{(l)} m \Big)^{-\alpha} L \Big(\log^{(l)} m\Big)$ for large $m$, where $\log^{(l)} m = \log \ldots \log m$. The proof ismainly based on Tauberian theorem. Using this result we also analyze theasymptotic behaviour of the first passage time $T_n$ of the state $n \in\mathbb{Z}$ by the walker in a neighborhood random walk in random environmentcreated by independent copies $(A_i : i \in \mathbb{Z})$ of $(0,1)$-valuedrandom variable $A$. This version differs from the final version as it containsan alternative proof for the tail behavior for generation sizes which is notvery sharp (lacks constant) but completely avoids arguments based on Tauberiantheorem. This proof may be of an independent interest.

Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching