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Growth rate of an endomorphism of a group

Growth rate of an endomorphism of a group Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map ƒ : M ↦↦ M on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient. We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

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Publisher
de Gruyter
Copyright
© de Gruyter 2011
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/GCC.2011.011
Publisher site
See Article on Publisher Site

Abstract

Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map ƒ : M ↦↦ M on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient. We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.

Journal

Groups - Complexity - Cryptologyde Gruyter

Published: Dec 1, 2011

Keywords: Growth rate; endomorphism; nilpotent groups; polycyclic groups

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