Compute δ(G) for the classical groups and their principal congruence subgroups

Compute the exact value of δ(G), the minimal positive integer δ such that G admits a faithful level-transitive self-similar action on a regular rooted δ-ary tree (equivalently, the minimal index of a simple virtual endomorphism), for each group G among SL_{l+1}(R), Sp_{2l}(R), SO_{2l}(R), and SO_{2l+1}(R), and for each of their m-th principal congruence subgroups (with m ≥ e), where R is the ring of integers of a p-adic field K and l ≥ 1 (with l ≥ 2 in the even-orthogonal case).

Background

The paper studies self-similarity of classical p-adic analytic groups and their associated Lie lattices. For these groups, the authors prove existence of self-similar actions and provide explicit self-similarity indices for large classes of open subgroups via virtual endomorphisms (Theorems A, B, G) and compute indices of principal congruence subgroups (Theorem D).

They define δ(G) as the minimal index δ for which a group G is self-similar of index δ, and in Corollary E provide upper bounds for δ(G) for the classical groups SL, Sp, and SO over the ring of integers R of a p-adic field and for their principal congruence subgroups. However, the exact minimal values δ(G) are not determined, leading to the explicit problem below.