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A group is said to be self-similar provided it admits a faithful state-closed representation on some regular m -tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level of the tree. A standard approach for constructing a transitive self-similar representation of a group has been by way of a single virtual endomorphism of the group in question. Recently, it was shown that this approach when applied to the restricted wreath product Z ≀ Z could not produce a faithful transitive self-similar representations for any m ≥ 2 (see [5]). In this work we study state-closed representations without assuming the transitivity condition. This general action is translated into a set of virtual endomorphisms corresponding to the different orbits of the action on the first level of the tree. In this manner, we produce faithful self-similar representations, some of which are also finite-state, for a number of groups such as Z ω , Z ≀ Z and (Z ≀ Z) ≀ C 2. [ABSTRACT FROM AUTHOR]

Abstract Let G be a group. The orbits of the natural action of Aut (G) on G are called "automorphism orbits" of G , and the number of automorphism orbits of G is denoted by ω (G). In this paper we prove that if G is an FC-group with finitely many automorphism orbits, then the derived subgroup G ′ is finite and G admits a decomposition G = Tor (G) × D , where Tor (G) is the torsion subgroup of G and D is a divisible characteristic subgroup of Z (G). We also show that if G is an infinite FC-group with ω (G) ⩽ 8 , then either G is soluble or G ≅ A 5 × H , where H is an infinite abelian group with ω (H) = 2. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits. [ABSTRACT FROM AUTHOR]

Let G p , d be the restricted wreath product C p w r C d where C p is a cyclic group of order a prime p and C d a free abelian group of finite rank d . We study the existence of faithful transitive state-closed ( fsc ) representations of G p , d on the rooted m -ary tree for some finite m . The group G 2 , 1 , known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for d ≥ 2 there are no fsc representations of G p , d on the p -adic tree. We describe all fsc representations of G = G p , 1 on the p -adic tree obtained via virtual endomorphisms, where the first level stabilizer of the image of G contains its commutator subgroup. Furthermore, for d ≥ 2 , we construct fsc representations of G p , d on the p 2 -adic tree and exhibit concretely the representation of G 2 , 2 on the 4-tree as a finite-state automaton group. [ABSTRACT FROM AUTHOR]