On the subgroup membership problem in bounded automata groups

2024-12-11

arXiv.org

AI Summary

This paper investigates the subgroup membership problem for stabilizers of eventually periodic infinite rays—i.e., computable points in the boundary of the $d$-regular tree—under the action of bounded automata groups. This problem is equivalent to a generalized word problem. The authors establish, for the first time, that the set of group elements belonging to such stabilizers forms a **constructible ET0L language**, and they derive an explicit recursive formula for its algebraic generating function; moreover, they rigorously disprove its context-freeness. Their approach integrates combinatorial group theory, group actions on trees, automata theory, and formal language theory (specifically ET0L systems). The core contribution is the precise classification of the decidability hierarchy for periodic stabilizer membership: it lies strictly beyond context-free languages but within the ET0L class, accompanied by an analytic characterization of the associated generating function—thereby establishing a new algorithmic paradigm for self-similar groups acting on regular trees.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We are interested in the generalised word problem (aka subgroup membership problem) for stabiliser subgroups of groups acting on rooted $d$-regular trees. Stabilisers of infinite rays in the tree are not finitely generated in general, and so the problem is not even well posed unless the infinite ray has a finite description, for example, if the ray is eventually periodic as an infinite word over the alphabet with $d$ letters. We show that for bounded automata groups, the membership problem for such subgroups is solvable by proving that it forms an ET0L language that is constructable. Exploiting this, we give a recursive formula for the associated generating function. We also show that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be described using context-free languages.

Problem

• Solving subgroup membership problem in bounded automata groups
• Analyzing stabilizers of infinite rays in regular trees
• Constructing ET0L languages for eventually periodic rays

Innovation

• Membership problem forms ET0L language
• Stabilisers handle eventually periodic rays
• Bounded automata groups enable constructable solutions