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GROWTHS OF ENDOMORPHISMS OF FINITELY GENERATED SEMIGROUPS

Part of: Special aspects of infinite or finite groups Semigroups

Published online by Cambridge University Press:  08 July 2016

ALAN J. CAIN Open the ORCID record for ALAN J. CAIN [Opens in a new window]  and
VICTOR MALTCEV
ALAN J. CAIN*
Affiliation:
Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal email a.cain@fct.unl.pt
VICTOR MALTCEV
Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa 32000, Israel email victor.maltcev@gmail.com
*
✉a.cain@fct.unl.pt
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Abstract

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This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

Keywords

growthfinitely generated semigroupsendomorphismCayley graph

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20M05: Free semigroups, generators and relations, word problems 20M99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 2 , April 2017 , pp. 163 - 184
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações). The first author was supported by an Investigador FCT research fellowship (IF/01622/2013/CP1161/CT0001).

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