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Groups with ET0L co-word problem

Raad Al Kohli, Derek F. Holt, Sarah Rees

Abstract

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.

Groups with ET0L co-word problem

Abstract

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.
Paper Structure (6 sections, 7 theorems, 2 equations)

This paper contains 6 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. Virtually free groups
  4. Bounded automata groups
  5. Higman--Thompson groups
  6. Closure properties of special co- et0l groups

Key Result

Lemma 2.1

Let $G = \langle A \rangle$ be a group, $\mathcal{M}$ a special cspda with $\mathcal{L}(\mathcal{M}) = \mathsf{coWP}(G,A)$, $w \in \mathcal{L}(\mathcal{M})$, and $n \in {\mathbb Z}$ with $n>0$. Then there is an entry configuration $C$ of $\mathcal{M}$ for which $C^w$ is accepting, and such that $C^v

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • proof
  • ...and 2 more