# Groups with Context‐Free Co‐Word Problem

@article{Holt2005GroupsWC, title={Groups with Context‐Free Co‐Word Problem}, author={Derek F. Holt and Sarah Rees and Claas E. R{\"o}ver and Richard M. Thomas}, journal={Journal of the London Mathematical Society}, year={2005}, volume={71} }

The class of co‐context‐free groups is studied. A co‐context‐free group is defined as one whose co‐word problem (the complement of its word problem) is context‐free. This class is larger than the subclass of context‐free groups, being closed under the taking of finite direct products, restricted standard wreath products with context‐free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co‐context‐free groups are known. It is proved that…

## 58 Citations

### The co‐word problem for the Higman‐Thompson group is context‐free

- Mathematics
- 2007

The co‐word problem of a group G generated by a set X is defined as the set of words in X which do not represent 1 in G. We introduce a new method to show that a permutation group has context‐free…

### LOCAL SIMILARITY GROUPS WITH CONTEXT-FREE CO-WORD PROBLEM

- Mathematics
- 2014

Let G be a group, and let S be a finite subset of G that generates G as a monoid. The co-word problem is the collection of words in the free monoid S ∗ that represent non-trivial elements of G. A…

### Groups with poly-context-free word problem

- MathematicsGroups Complex. Cryptol.
- 2014

It is shown that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds is proved.

### Groups with Indexed Co-word Problem

- MathematicsInt. J. Algebra Comput.
- 2006

It is shown that all Higman–Thompson groups and a large class of tree automorphism groups defined by finite automata are co-indexed groups, including the Grigorchuk 2-group and the Gupta–Sidki 3-group.

### Context-Freeness of Higman-Thompson group's co-word problem

- Mathematics
- 2008

AbstractThe co-word problem of a group G generated by a set X is deﬁned asthe set of words in X which do not represent 1 in G. We introduce a newmethod to decide if a permutation group has…

### Non-finitely Generated Maximal Subgroups of Context-free Monoids

- MathematicsJournal of Algebra
- 2022

### Word Problem Languages for Free Inverse Monoids

- MathematicsDCFS
- 2018

It is shown that no free inverse monoid has context-free word problem, and that the word problem of the free inversemonoid of rank $1$ is both $2$-context-free (an intersection of two context- free languages) and ET0L.

### Semigroups with a Context-Free Word Problem

- MathematicsDevelopments in Language Theory
- 2012

The depth of the Muller-Schupp result and its reliance on the geometrical structure of Cayley graphs of groups suggests that a generalization to semigroups could be very hard to obtain, but some results are able to prove about this intriguing class of semig groups.

### Church-Rosser Groups and Growing Context-Sensitive Groups

- Mathematics, Computer ScienceJ. Autom. Lang. Comb.
- 2008

The free abelian group of rank 2 is a non-context-free Church-Rosser group, and it is shown that there are co- context-free groups that are not growing context-sensitive.

### Context-free word problem semigroups

- MathematicsDLT
- 2019

Some examples are exhibited to clarify the relationship between semigoups and monoids and their connection with the notions of word-hyperbolicity and automaticity and whether these classes are closed under applying certain semigroup constructions, including direct products and free products.

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