Spinal Hanoi Towers Groups

We introduce and study \emph{spinal Hanoi towers groups}, a family of groups acting on the $d$-adic tree that contains both the classical Hanoi towers group $\mathcal{H}^{(3)}$ and Skipper's generalizations as extreme cases. Each group is generated by $d$ automorphisms $a_1,\dots,a_d$, where $a_i$ has a unique non-trivial section, equal to $a_i$ itself, at the $i$-th coordinate, and root permutation $\sigma_i$ fixing $i$. The entire construction is thus encoded by the finite permutation group $P=\langle\sigma_1,\dots,\sigma_d\rangle\leq\mathrm{Sym}(d)$, and we develop a dictionary between the two: $G$ is fractal and level transitive if and only if $P$ is transitive; every group in the family is amenable and contracting, with explicit nucleus and a word problem solved by a length-halving recursion on syllables; and the abelianizations of $G$ and $P$ together control the first level stabilizer.
The branch structure of the family is governed by the subgroup $J\leq\text{Aut}(\mathcal{T}_d)$, generated by the automorphisms with exactly two non-trivial sections, occupied by an element $h\in G$ and its inverse. We give a criterion for the containment $J\leq G$ that can be verified in an explicit finite quotient, and we prove that whenever it holds, a level transitive spinal Hanoi towers group is strongly fractal, regular branch over its commutator subgroup, has explicitly described rigid stabilizers at every level, and is just infinite. As an application, we show that the groups of type $(d,m)$, whose root permutations are $m$-cycles, satisfy $J\leq G$ for all $d\geq 4$ and are therefore just infinite, in contrast with the classical case of the Hanoi towers group on three pegs.