Abstract. L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the BridsonGilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth.
We show that the class of $\mathcal{C}$-hereditarily conjugacy separable
groups is closed under taking arbitrary graph products whenever the class
$\mathcal{C}$ is an extension closed variety of finite groups. As a consequence
we show that the class of $\mathcal{C}$-conjugacy separable groups is closed
under taking arbitrary graph products. In particular, we show that right angled
Coxeter groups are hereditarily conjugacy separable and 2-hereditarily
conjugacy separable, and we show that infinitely generated right angled Artin
groups are hereditarily conjugacy separable and $p$-hereditarily conjugacy
separable for every prime number $p$.Comment: 40 page
We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products.Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups -which are called p-isolated -are closed in the pro-p topology of the graph product. In particular, we show that every pisolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology, and we fully characterise such subgroups.
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