On self-similarity of p-adic analytic pro-p groups of small dimension

Abstract Given a torsion-free p-adic analytic pro-p group G with dim ( G ) p , we show that the self-similar actions of G on regular rooted trees can be studied through the virtual endomorphisms of the associated Z p -Lie lattice. We explicitly classify 3-dimensional unsolvable Z p -Lie lattices for p odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free p-adic analytic pro-p groups for p ⩾ 5 , and to determine which of them admit a faithful self-similar action on a p-ary tree. In particular, we show that no open subgroup of S L 1 1 ( Δ p ) admits such an action. On the other hand, we prove that all the open subgroups of S L 2 △ ( Z p ) admit faithful self-similar actions on regular rooted trees.