Covering number for reflections in trees
| Autor: | Humberto Luiz Talpo, Marcelo Firer |
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| Rok vydání: | 2008 |
| Předmět: | |
| Zdroj: | Journal of Group Theory. 11 |
| ISSN: | 1435-4446 1433-5883 |
| DOI: | 10.1515/jgt.2008.055 |
| Popis: | We define a reflection in a tree as an involutive automorphism whose set of fixed points is a geodesic and prove that, for the case of a homogeneous tree of even degree, the group of even automorphisms may be covered by at most 11 reflections. Reflections are defined as involutive automorphisms having a geodesic as set of fixed points. In a previous work ([6]) we studied the structure of reflections in an homogeneous tree Γ of degree k ≡ 0mod 4. We considered the group 〈R〉 generated by the set of all reflections R and the (index two) subgroup Aut (Γ) ⊂ Aut (Γ) consisting of automorphisms with even displacement function and proved that the topological closure of 〈R〉 is Aut (Γ), that is, given φ ∈ Aut (Γ) there is a sequence (φn)n=1 with φn ∈ 〈R〉 and a sequence of subsets Ai ⊂ Γ with An ⊂ An+1 and Γ = ∪n=1An such that φn coincides with φ on An, that is, φn|An = φ|An . The proof given is constructive and actually each φn is the product of n reflections, so we could not say that Aut (Γ) is finitely generated by R. In this work we prove with simple arguments that Aut (Γ) is finitely generated by R (Proposition 2) and go further, proving that every φ ∈ Aut (Γ) may be expressed as the product of at most eleven reflections (Theorem 12). 1 Basic Concepts The free monoid X∗ of words over the alphabet X = {0, ..., n− 1} ordered by the prefix relation has a n-regular rooted tree structure in which the empty word is the root and the words of length l constitute the level l in the tree. Denote this n-regular rooted tree by T . If we consider two copies T ′ and T ′′ of the n-regular rooted tree T and add a single edge, connecting the root of T ′ to the root of T ′′, we get a k-homogeneous tree Γ, that is, ∗Author supported by CAPES. |
| Databáze: | OpenAIRE |
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