Free-by-finite pro-p groups and integral p-adic representations

Abstract

Let F be a free pro-p group of finite rank n and \({C_{p^r}}\) a cyclic group of order pr. In this work we classify p-adic representations \({ C_{p^r}\longrightarrow GL_n(\mathbb{Z}_{p})}\) that can be obtained as a composite of an embedding \({C_{p^r}\longrightarrow {\rm Aut}(F)}\) with the natural epimorphism \({{\rm Aut}(F)\longrightarrow GL_n(\mathbb{Z}_{p})}\) .