Groups with ET0L co-word problem

We study groups whose co-word problems are ET0L languages, which we call coET0L groups, using an automaton based model due to van Leeuwen, and recently studied by Bishop and Elder. In particular we prove a number of closure results for the class of groups with co-word problems in a subclass of `special’ ET0L languages; that class of groups contains all groups that we know at the time of writing to be co-ET0L, including all groups that were proved by Holt and Röver to be stack groups, and hence co-indexed. It includes virtually free groups, bounded automata groups, and the Higman-Thompson groups, together with groups constructed from those using finitely generated subgroups, finite extension, free and direct products, and by taking the restricted standard wreath product of a co-\E group by a finitely generated virtually free top group.

💡 Research Summary

The paper introduces the notion of a “co‑ET0L group”, namely a finitely generated group whose co‑word problem (the set of words over a fixed generating set that do not represent the identity) belongs to the class of ET0L languages. The authors adopt the automaton model of van Leeuwen, later refined by Bishop and Elder, called a check‑stack push‑down automaton (cspda). By imposing seven structural constraints (P1–P7) on a cspda they define a “special cspda”, whose accepted languages are exactly the “special ET0L languages”. The main thrust of the paper is to show that a large and natural collection of groups falls into this class and that the class is closed under several standard group‑theoretic constructions.

Key technical framework.
A cspda has two stacks: a check‑stack (Δ) and a push‑down stack (Γ). Computation proceeds in two phases. In the first phase the machine may write symbols on the check‑stack but cannot read the input; the set of possible check‑stack strings is regular. In the second phase the machine reads the input while the check‑stack is read‑only; each push onto the push‑down stack forces the read‑head to move one cell up the check‑stack, and each pop forces it to move down. The constraints (P1)–(P7) enforce deterministic behaviour, forbid infinite sequences of non‑reading moves, prescribe a unique bottom and top symbol for each stack, and require that every non‑entry reading state be accepting. Crucially, (P6) guarantees that if a prefix of an input word is not in the language, the automaton either stays in the same configuration or fails immediately; this “ignoring the word problem” property mirrors the notion of a stack group in Holt‑Röver’s work.

Main existence results.

1. Virtually free groups. Using the classical construction of a deterministic push‑down automaton for the word problem of a virtually free group (Muller–Schupp), the authors augment it with a check‑stack of sufficient length and obtain a special cspda that accepts exactly the complement of the word problem. Hence every finitely generated virtually free group is a special co‑ET0L group.

2. Bounded automata groups. These include the Grigorchuk group, Gupta–Sidki groups, and many basilica‑type groups. Earlier work (Holt‑Röver) showed they are co‑indexed (their co‑word problems are indexed languages). Bishop and Elder later proved they are co‑ET0L. The present paper refines that result by showing that the automata can be arranged to satisfy the seven constraints, thereby placing all finitely generated bounded automata groups in the special co‑ET0L class.

3. Higman–Thompson groups. Although previously known to have co‑context‑free word problems, the authors construct a special cspda that records the traversal of the underlying binary (or n‑ary) tree on the check‑stack while using the push‑down stack to keep track of the current node. This yields a special ET0L description of the co‑word problem.

Closure properties.
Theorem 6.1 establishes that the property of being a special co‑ET0L group does not depend on the chosen finite inverse‑closed generating set. Subsequent theorems prove that the class is closed under:

• Taking finitely generated subgroups (Lemma 2.1 provides a technical tool to avoid failure on short words).
• Finite direct products and finite extensions.
• Free products with finitely many free factors.
• Restricted standard wreath products where the top group is finitely generated and virtually free.

The proofs repeatedly exploit (P6) to control how the automaton reacts to words outside the language, and they use padding symbols on the check‑stack to guarantee that any generating set can be accommodated by enlarging the initial check‑stack content.

Independence of generating set.
By inserting a sufficiently long block of padding symbols ‘p’ at the start of the check‑stack, the authors show that any change of generators can be simulated without altering the accepted language. This mirrors the standard argument that the co‑word problem is a language property independent of the generating set, once the automaton is allowed to ignore irrelevant prefixes.

Significance.
The work lifts several previously known results from the indexed‑language level to the stricter ET0L level, thereby sharpening our understanding of the computational complexity of co‑word problems. The introduction of the special cspda model provides a concrete, automata‑theoretic tool for constructing ET0L recognisers for group‑theoretic languages. Moreover, the extensive closure results suggest that the special co‑ET0L class is robust under natural group constructions, making it a promising candidate for a “natural” complexity class in geometric group theory, analogous to the role of context‑free languages for virtually free groups (Muller–Schupp) and indexed languages for bounded automata groups (Holt‑Röver).

In summary, the paper systematically builds a bridge between ET0L formal language theory and the algebraic study of groups, delivering explicit automaton constructions for a broad spectrum of groups and proving that the resulting class enjoys strong algebraic closure properties. This advances the program of classifying groups by the formal complexity of their word‑related decision problems.