Asymptotic invariants of residually finite just infinite groups

Recently, Eduard Schesler and the second author constructed examples of finitely generated residually finite, hereditarily just infinite groups with positive first \(L^2\) $L^2$ -Betti number. In contrast to their result, we show that a finitely generated residually-p just infinite group has trivial first \(L^2\) $L^2$ -Betti number. Moreover, we prove that the normal homology rank gradient of a finitely generated, residually finite, just infinite group vanishes.