Automorphism Groups of Semi-Homogeneous Trees Humberto Luiz Talpo and Marcelo Firer Abstract. In this work, we prove that automorphism groups of semi-homogeneous trees are complete and ambivalent. There are many works studying the structure of homogeneous trees (trees in which all vertices have the same degree) or semi-homogeneous trees (trees in which vertices have two different degrees and the vertices x and y have the same degree iff d (x, y) ≡ 0 mod 2). J. Tits [5] proved that automorphism groups (without in- version) of locally finite homogeneous or semi-homogeneous trees are simple. D. V. Znoiko [6] proved that automorphism groups of homogeneous trees (of degree n> 2) are complete. P.W. Gawron et al [1] gave a full description of conjugacy classes in the automorphism group of trees and proved that those groups are ambiva- lent. In this work, we prove that automorphism groups of semi-homogeneous trees are complete and using different tools and techniques to show that automorphism groups of semi-homogeneous trees are ambivalent and moreover, any automorphism is conjugated to its inverse by an involution. 1. Preliminaries Let Γ be a locally finite simplicial tree. We denote the set of vertices of Γ by VΓ and the set of edges by Γ by E(Γ). As we are concerned with nondirected trees only, we denote by {x, y} the edge joining the vertices x, y ∈ V(Γ). If {x, y}∈ E(Γ) then the vertices x, y are called adjacent. Every tree Γ may be equipped with a natural metric d. For any two vertices x, y ∈ V(Γ) we define d(x, y) to be the minimal number of edges in an edge-path from x to y. If we endow each edge with the metric of the unit interval [0, 1] ⊂ R, then d naturally extend to a metric on Γ and, with such a metric, Γ is a CAT(0) we have unicity of minimal path (geodesic) connecting a pair of given points. We denote by S (x, t) the usual metric sphere on VΓ, centered at x with radius t and B (x, t) denote the usual metric ball, centered at x with radius t. By an automorphism of Γ we mean a bijective isometry of (Γ,d) which takes vertices to vertices (and therefore edges to edges). We denote by Aut(Γ) the group of automorphisms of Γ. The degree of a vertex x ∈ V(Γ), denoted by degree(x), is the number of vertices adjacent to vertex x, i.e., degree(x)= |S (x, 1)|. For k ≥ 2 we define a 1991 Mathematics Subject Classification. [2000]Primary 20E08, 20E36; Secondary 05C05. Key words and phrases. Semi-homogeneous Trees, Automorphism Groups. 1