A finitely generated group is called a Church-Rosser group (growing context-sensitive group) if it admits a finitely generated presentation for which the word problem is a Church-Rosser (growing context-sensitive) language.
Although the Church-Rosser languages are incomparable to the context-free languages under set inclusion, they strictly contain the class of deterministic context-free languages.
As each context-free group language is actually deterministic context-free, it follows that all context-free groups are Church-Rosser groups. As the free abelian group of rank 2 is a non-context-free Church-Rosser group, this inclusion is proper.
On the other hand, we show that there are co-context-free groups that are not growing context-sensitive. Also some closure and non-closure properties are established for the classes of Church-Rosser and growing context-sensitive groups.
More generally, we also establish some new characterizations and closure properties for the classes of Church-Rosser and growing context-sensitive languages.