Abstract
This paper surveys some classical results about growth in finitely generated semigroups and applies results from additive number theory to construct families of finitely generated linear semigroups with intermediate growth.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
V.V. Beljaev, N.F. Sesekin, V.I. Trofimov, Growth functions of semigroups and loops. Ural. Gos. Univ. Mat. Zap. 10(3) (1977), Issled. po Sovremen. Algebre, 3–8, 215. MR MR0480783 (58 #933)
M. Gromov, Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)
A.G. KhovanskiÄ, The Newton polytope, the Hilbert polynomial and sums of finite sets. Funktsional. Anal. i Prilozhen. 26(4), 57–63, 96 (1992)
A.G. KhovanskiÄ, Sums of finite sets, orbits of commutative semigroups and Hilbert functions. Funktsional. Anal. i Prilozhen. 29(2), 36–50, 95 (1995)
B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth. J. Amer. Math. Soc. 23, 815–829 (2010)
A.A. Lavrik-Männlin, On some semigroups of intermediate growth. Int. J. Algebra Comput. 11(5), 565–580 (2001)
J. Milnor, A note on curvature and fundamental group. J. Differ. Geom. 2, 1–7 (1968)
M.B. Nathanson, Number theory and semigroups of intermediate growth. Am. Math. Monthly 106(7), 666–669 (1999)
M.B. Nathanson, Elementary Methods in Number Theory. Graduate Texts in Mathematics, vol. 195 (Springer, New York, 2000)
M.B. Nathanson, Growth of sumsets in abelian semigroups. Semigroup Forum 61(1), 149–153 (2000)
M.B. Nathanson, Partitions with parts in a finite set. Proc. Am. Math. Soc. 128(5), 1269–1273 (2000)
M.B. Nathanson, I.Z. Ruzsa, Polynomial growth of sumsets in abelian semigroups. J. Théor. Nombres Bordeaux 14(2), 553–560 (2002)
J. Okniński, Semigroups of Matrices. Series in Algebra, vol. 6 (World Scientific Publishing Co. Inc., River Edge, 1998)
Y. Shalom, T. Tao, A finitary version of Gromov’s polynomial growth theorem. Geom. Funct. Anal. 20, 1502–1547 (2010)
T. Tao, A new proof of Gromov’s theorem (2010), http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem
L. van den Dries, A.J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89(2), 349–374 (1984)
Acknowledgements
This work was supported in part by a grant from the PSC-CUNY Research Award Program.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this paper
Cite this paper
Nathanson, M.B. (2014). Additive Number Theory and Linear Semigroups with Intermediate Growth. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_13
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1601-6_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1600-9
Online ISBN: 978-1-4939-1601-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)