Groups, Geometry, and Dynamics

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Volume 1, Issue 1, 2007, pp. 21–46
DOI: 10.4171/GGD/2

Published online: 2007-03-31

Virtual endomorphisms of nilpotent groups

Adilson A. Berlatto[1] and Said N. Sidki[2]

(1) Universidade Federal de Mato Grosso, Pontal Do Araguaia, Brazil
(2) Universidade de Brasília, Brazil

A virtual endomorphism of a group G is a homomorphism f : HG where H is a subgroup of G of finite index m. The triple (G,H,f) produces a state-closed (or, self-similar) representation φ of G on the 1-rooted m-ary tree. This paper is a study of properties of the image Gφ when G is nilpotent. In particular, it is shown that if G is finitely generated, torsion-free and nilpotent then Gφ has solvability degree bounded above by the number of prime divisors of m.

Keywords: Virtual endomorphisms, nilpotent groups, automorphisms of trees, state-closed representations

Berlatto Adilson, Sidki Said: Virtual endomorphisms of nilpotent groups. Groups Geom. Dyn. 1 (2007), 21-46. doi: 10.4171/GGD/2