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On the ET0L subgroup membership problem in bounded automata groups

Alex Bishop, Daniele D'Angeli, Francesco Matucci, Tatiana Nagnibeda, Davide Perego, Emanuele Rodaro

TL;DR

This work studies the subgroup membership problem for groups acting on rooted $d$-regular trees, focusing on stabilisers of infinite rays. For bounded automata groups and rays that are computable and eventually periodic, it proves that the corresponding membership language $WP(G,X,\mathrm{Stab}(\eta))$ is an ET0L language, and it provides constructible, unambiguous limiting ET0L grammars that describe both the language and its complement. It further shows that this result is optimal in general, since the language need not be context-free in many prominent examples, yet it yields computable generating functions (Green functions) on the associated Schreier graphs. The paper also develops a framework of unambiguous limiting ET0L grammars, closure under string transductions, and a mechanism to extract uniform, computable descriptions across all bounded automata groups and eventual-ray inputs. Collectively, these results bridge formal language theory with geometric group theory, enabling decidability and generating-function analysis for a broad class of subgroup membership problems in automata groups.

Abstract

We are interested in the subgroup membership problem in groups acting on rooted $d$-regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.

On the ET0L subgroup membership problem in bounded automata groups

TL;DR

This work studies the subgroup membership problem for groups acting on rooted -regular trees, focusing on stabilisers of infinite rays. For bounded automata groups and rays that are computable and eventually periodic, it proves that the corresponding membership language is an ET0L language, and it provides constructible, unambiguous limiting ET0L grammars that describe both the language and its complement. It further shows that this result is optimal in general, since the language need not be context-free in many prominent examples, yet it yields computable generating functions (Green functions) on the associated Schreier graphs. The paper also develops a framework of unambiguous limiting ET0L grammars, closure under string transductions, and a mechanism to extract uniform, computable descriptions across all bounded automata groups and eventual-ray inputs. Collectively, these results bridge formal language theory with geometric group theory, enabling decidability and generating-function analysis for a broad class of subgroup membership problems in automata groups.

Abstract

We are interested in the subgroup membership problem in groups acting on rooted -regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.

Paper Structure

This paper contains 25 sections, 23 theorems, 86 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Bounded Automata Groups
  3. ET0L Languages
  4. Unambiguous Limiting Grammars
  5. Closure under mapping by string transducer
  6. Generating Functions
  7. Main theorem
  8. Nonterminals and starting symbols
  9. Nonterminals with restrictions on result of a word action
  10. Additional nonterminals for $E'$
  11. Intuition of the starting symbols
  12. Computability of nonterminals
  13. Initialisation map: $\tau_\mathrm{init}$
  14. Initialising the grammar $E$
  15. Initialising the grammar $E'$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that we are given a finite symmetric set $X$ of bounded automaton automorphisms acting on the tree $\mathcal{T}_d = C^\ast$, and words $a,b\in C^*$. Then, we can effectively compute---uniformly over all $X$, $a$ and $b$---an ET0L grammar which generates the language $\mathsf{WP}(G,X,\mathrm{

Figures (6)

  • Figure 1: A labelling of the vertices of $\mathcal{T}_d$.
  • Figure 2: Examples of finitary automorphisms $a,b\in\mathrm{Fin}(\mathcal{T}_d)$.
  • Figure 3: Examples of directed automorphisms $x,y,z \in \mathrm{Dir}(\mathcal{T}_2)$.
  • Figure 4: Derivation tree for $a^2 b a b a b$ labelled by $\alpha\beta\alpha\alpha\gamma$.
  • Figure 5: Automaton of the infinite dihedral group.
  • ...and 1 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.1
  • Proposition 1.0
  • Proposition 1.0
  • Proposition 1.0
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 42 more