By Mugnai D.

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R, such that the following conditions are met: 1. T = rj=1 {ǫj (t) : t ∈ [0, 1]}. 2. If ǫj (t) = ǫk (s) and j = k then t, s ∈ {0, 1}. 3. T is simply connected. In this paper we will be concerned only with ﬁnite trees and we will always mean a ﬁnite tree when we speak of a tree. The set V = {ǫ1 (0), ǫ1 (1), . . , ǫr (0), ǫr (1)} has precisely r + 1 elements called vertices of the tree. The homeomorphisms ǫj are called edges of the tree. We say a vertex v belongs to an edge ǫj or that edge ǫj is attached to v if v = ǫj (0), the initial vertex of ǫj , or v = ǫj (1), the terminal vertex of ǫj .

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331(9):663–668, 2000. I. Grigorchuk and A. Zuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata, 87:209–244, 2001. [41] G. Grimmett. Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1999. ´ [42] M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. , 53:53–73, 1981. [43] M. A. Shubin. von Neumann spectra near zero. Geom. Funct. , 1(4):375–404, 1991. W. Kantelhardt and A.

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