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Groups Complexity Cryptology

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Growth rate of an endomorphism of a group

Kenneth J. Falconer
  • Mathematical Institute, University of St Andrews, Fife KY16 9SS, Scotland.
  • :
/ Benjamin Fine
  • Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA.
  • :
/ Delaram Kahrobaei
  • Doctoral Program in Computer Science, CUNY Graduate Center, City University of New York, New York, NY 10016, USA.
  • :
Published Online: 2011-11-03 | DOI: https://doi.org/10.1515/gcc.2011.011


Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map ƒ : MM 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.

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

Received: 2010-12-28

Revised: 2011-03-18

Published Online: 2011-11-03

Published in Print: 2011-12-01

Citation Information: Groups – Complexity – Cryptology. Volume 3, Issue 2, Pages 285–300, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc.2011.011, November 2011

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