The word problem in Hanoi Towers groups

TL;DR

This paper demonstrates that the word problem in Hanoi Towers groups can be solved efficiently, with the complexity bounded by a poly-logarithmic function, improving understanding of their computational properties.

Contribution

It establishes a poly-logarithmic upper bound on the depth of elements in Hanoi Towers groups, leading to a subexponential time solution for the word problem.

Findings

Depth of elements is bounded by O(log^{m-2} n)

Word problem solvable in subexponential time exp(O(log^{m-2} n))

Provides new bounds on computational complexity in Hanoi Towers groups

Abstract

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 $\exp(O(\log^{m-2} n))$.