Morphically primitive words

In the present paper, we introduce an alternative notion of the primitivity of words, that–unlike the standard understanding of this term–is not based on the power (and, hence, the concatenation) of words, but on morphisms. For any alphabet Σ, we call a word wΣ* morphically imprimitive provided that there are a shorter word v and morphisms h,h′:Σ*→Σ* satisfying h(v)=w and h′(w)=v, and we say that w is morphically primitive otherwise. We explain why this is a well-chosen terminology, we demonstrate that morphic (im-) primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms, and we study a number of fundamental properties of the concepts under consideration

Complexity of testing morphic primitivity

We analyze the algorithm in [Holub, 2009], which decides whether a given word is a fixed point of a nontrivial morphism. We show that it can be implemented to have complexity in O(mn), where n is the length of the word and m the size of the alphabet

On Quasiperiodic Morphisms

Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.Comment: 12 page

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat

Uniform symplicity of groups with proximal action

We prove that groups acting boundedly and order-primitively on linear orders or acting extremly proximality on a Cantor set (the class including various Higman-Thomson groups and Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms) are uniformly simple. Explicit bounds are provided.Comment: 23 pages, appendix by Nir Lazarovich, corrected versio

On almost cylindrical languages and the decidability of the D0L and PWD0L primitivity problems

AbstractPrimitive words and their properties have always been of fundamental importance in the study of formal language theory. Head and Lando in Periodic D0L Languages proposed the idea of deciding whether or not a given D0L language has the property that every word in it is a primitive word. After reducing the general problem to the case in which h is injective, it will be shown that primitivity is decidable when ((A)h)∗ is an almost cylindrical set. Moreover, in this case, it is shown that the set of words which generate primitive sequences (given a particular D0L scheme) is an algorithmically constructible context-sensitive language. An undecidability result for the PWD0L primitivity problem and decidability results for cases of the RWD0L primitivity problem are also given