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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 [email protected]
VICTOR MALTCEV
Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa 32000, Israel email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

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).

References

Baader, F. and Nipkow, T., Term Rewriting and All That (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Bestvina, M., Feighn, M. and Handel, M., ‘The Tits alternative for Out(F n ). I. Dynamics of exponentially-growing automorphisms’, Ann. of Math. (2) 151(2) (2000), 517–623.Google Scholar
Book, R. V. and Otto, F., String-Rewriting Systems, Texts and Monographs in Computer Science (Springer, New York, 1993).Google Scholar
Bowen, R., ‘Entropy and the fundamental group’, in: The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State University, Fargo, ND, 1977), Lecture Notes in Mathematics, 668 (Springer, Berlin, 1978), 21–29.CrossRefGoogle Scholar
Cain, A. J., ‘Presentations for subsemigroups of groups’, PhD Thesis, University of St Andrews, 2005.Google Scholar
Cain, A. J., ‘Hyperbolicity of monoids presented by confluent monadic rewriting systems’, Beiträge Algebra Geom. (2) 54(10) (2013), 593–608.Google Scholar
Cain, A. J., Gray, R. and Ruškuc, N., ‘Green index in semigroup theory: generators, presentations, and automatic structures’, Semigroup Forum 85(3) (2012), 448–476.Google Scholar
Cain, A. J. and Maltcev, V., ‘For a few elements more: a survey of finite Rees index’,arXiv:1307.8259.Google Scholar
Cain, A. J. and Maltcev, V., ‘Context-free rewriting systems and word-hyperbolic structures with uniqueness’, Int. J. Algebra Comput. 22(7) (2012).Google Scholar
Cain, A. J. and Maltcev, V., ‘Hopfian and co-hopfian subsemigroups and extensions’, Demonstratio Math. 47(4) (2014), 791–804.Google Scholar
Cain, A. J., Robertson, E. F. and Ruškuc, N., ‘Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness’, Math. Proc. Cambridge Philos. Soc. 141(1) (2006), 57–66.Google Scholar
Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. II, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1967).CrossRefGoogle Scholar
Dicks, W. and Ventura, E., ‘Irreducible automorphisms of growth rate one’, J. Pure Appl. Algebra 88(1–3) (1993), 51–62.Google Scholar
Dikranjan, D. and Giordano Bruno, A., ‘Discrete dynamical systems in group theory’, Note Mat. 33(1) (2013), 1–48.Google Scholar
Duncan, A. and Gilman, R. H., ‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136(3) (2004), 513–524.Google Scholar
Falconer, K. J., Fine, B. and Kahrobaei, D., ‘Growth rate of an endomorphism of a group’, Groups Complex. Cryptol. 3(2) (2011), 285–300.Google Scholar
Gray, R. and Kambites, M., ‘A Švarc–Milnor lemma for monoids acting by isometric embeddings’, Int. J. Algebra Comput. 21(7) (2011), 1135–1147.Google Scholar
Gray, R. and Kambites, M., ‘Groups acting on semimetric spaces and quasi-isometries of monoids’, Trans. Amer. Math. Soc. 365(2) (2013), 555–578.Google Scholar
Gray, R., Maltcev, V., Mitchell, J. D. and Ruškuc, N., ‘Ideals and finiteness conditions for subsemigroups’, Glasg. Math. J. 56 (2014), 65–86.Google Scholar
Gray, R. and Ruškuc, N., ‘Green index and finiteness conditions for semigroups’, J. Algebra 320(8) (2008), 3145–3164.Google Scholar
Gromov, M., ‘Hyperbolic groups’, in: Essays in Group Theory, Mathematical Sciences Research Institute Publications, 8 (ed. Gersten, S. M.) (Springer, New York, 1987), 75–263.CrossRefGoogle Scholar
Howie, J. M., Fundamentals of Semigroup Theory, London Mathematical Society Monographs (New Series), 12 (Clarendon Press–Oxford University Press, New York, 1995).Google Scholar
Jackson, D. A. and Kilibarda, V., ‘Ends for monoids and semigroups’, J. Aust. Math. Soc. 87(1) (2009), 101–127.Google Scholar
Kilibarda, V., Maltcev, V. and Craik, S., ‘Ends for subsemigroups of finite index’, Semigroup Forum 91(2) (2015), 401–414.CrossRefGoogle Scholar
Lallement, G., Semigroups and Combinatorial Applications (John Wiley, New York–Chichester–Brisbane, 1979).Google Scholar
Lawson, M. V., Inverse Semigroups: The Theory of Partial Symmetries (World Scientific, River Edge, NJ, 1998).Google Scholar
Levitt, G. and Lustig, M., ‘Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups’, Comment. Math. Helv. 75(3) (2000), 415–429.Google Scholar
Maltcev, V., Mitchell, J. D. and Ruškuc, N., ‘The Bergman property for semigroups’, J. Lond. Math. Soc. (2) 80(1) (2009), 212–232.Google Scholar
Maltcev, V. and Ruškuc, N., ‘On hopfian cofinite subsemigroups’, arXiv:1307.6929.Google Scholar
Muller, D. E. and Schupp, P. E., ‘Groups, the theory of ends, and context-free languages’, J. Comput. System Sci. 26(3) (1983), 295–310.Google Scholar
Myasnikov, A. G. and Shpilrain, V., ‘Some metric properties of automorphisms of groups’, J. Algebra 304(2) (2006), 782–792.Google Scholar
Robertson, E. F., Ruškuc, N. and Wiegold, J., ‘Generators and relations of direct products of semigroups’, Trans. Amer. Math. Soc. 350(7) (1998), 2665–2685.Google Scholar
Rosser, J. B. and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.Google Scholar
Silva, P. V. and Steinberg, B., ‘A geometric characterization of automatic monoids’, Q. J. Math. 55(3) (2004), 333–356.CrossRefGoogle Scholar