脊柱河内塔群
Spinal Hanoi Towers Groups
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中文总结 AI 辅助
研究脊柱河内塔群，它由\(d\)个自同构生成，与有限置换群\(P\)相关。建立两者对应关系，给出\(J\leq G\)的准则。证明满足该准则时，群具有强分形等性质，还表明根置换为\(m\)循环的\((d,m)\)型群满足条件且恰好无限。
中文摘要 AI 辅助
我们引入并研究了脊柱河内塔群，这是一族作用于\(d\)进树的群，其中经典河内塔群\(\mathcal{H}^{(3)}\)和斯基珀的推广作为极端情况包含在内。每个群由\(d\)个自同构\(a_1,\dots,a_d\)生成，其中\(a_i\)在第\(i\)个坐标处有唯一非平凡截面，等于其自身，且根置换\(\sigma_i\)固定\(i\)。整个构造由有限置换群\(P = \langle\sigma_1,\dots,\sigma_d\rangle\leq\mathrm{Sym}(d)\)编码。我们建立了两者之间的对应关系：\(G\)是分形且层级传递的当且仅当\(P\)是传递的；该族中的每个群都是顺从且收缩的，有明确的核，且通过对音节进行长度减半递归解决字问题；\(G\)和\(P\)的交换化共同控制第一层级稳定子。该族的分支结构由子群\(J\leq\text{Aut}(\mathcal{T}_d)\)控制，\(J\)由具有恰好两个非平凡截面的自同构生成，由\(G\)中的元素\(h\)及其逆占据。我们给出了\(J\leq G\)的包含准则，可在显式有限商中验证。我们证明，只要该准则成立，层级传递的脊柱河内塔群就是强分形的，在其换位子群上是正则分支的，在每个层级都有明确描述的刚性稳定子，并且是恰好无限的。作为应用，我们表明根置换为\(m\)循环的\((d,m)\)型群，对于所有\(d\geq 4\)都满足\(J\leq G\)，因此是恰好无限的，这与三钉河内塔群的经典情况形成对比。
英文摘要
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 $σ_i$ fixing $i$. The entire construction is thus encoded by the finite permutation group $P=\langleσ_1,\dots,σ_d\rangle\leq\mathrm{Sym}(d)$, 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.
