For each prime p and monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. In this setting, the first Grigorchuk group is the group G_{2,x^2+x+1}. We show that the closures of these groups are finitely constrained. This enables us to calculate their Hausdorff dimension, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1.
Further ``finiteness'' properties are also discussed (amenability, torsion, and intermediate growth).