Modified Hanoi Towers Groups and Limit Spaces

Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.

A constructive proof that the Hanoi towers group has non-trivial rigid kernel

In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel of the Hanoi towers group. We present a simpler proof that the rigid kernel is the Klein 4 group. In the course of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, recurrent, regular branch group.

Schreier spectrum of the Hanoi Towers group on three pegs

Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided.

The word problem in Hanoi Towers groups

We prove that elements of the Hanoi Towers groups $\mathcal{H}_m$ have depth bounded from above by a poly-logarithmic function $O(\log^{m-2} n)$, where $n$ is the length of an element. Therefore the word problem in groups $\mathcal{H}_m$ is solvable in subexponential time $\exp(O(\log^{m-2} n))$.

Twin Towers of Hanoi

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a …

脊柱河内塔群

Spinal Hanoi Towers Groups

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。