Semigroup Forum

, Volume 92, Issue 1, pp 23–44 | Cite as

Growth degree classification for finitely generated semigroups of integer matrices

  • Jason P. Bell
  • Michael Coons
  • Kevin G. Hare
Research Article


Let \({\mathcal {A}}\) be a finite set of \(d\times d\) matrices with integer entries and let \(m_n({\mathcal {A}})\) be the maximum norm of a product of \(n\) elements of \({\mathcal {A}}\). In this paper, we classify gaps in the growth of \(m_n({\mathcal {A}})\); specifically, we prove that \(\lim _{n\rightarrow \infty } \log m_n({\mathcal {A}})/\log n\in \mathbb {Z}_{\geqslant 0}\cup \{\infty \}.\) This has applications to the growth of regular sequences as defined by Allouche and Shallit.


Finitely generated semigroups Matrix semigroups Automatic sequences Regular sequences 



We thank Vladimir Protasov for bringing our attention to his current result with Jungers [22] as well as providing us with a preprint of that work. Research of J. P. Bell was supported by NSERC Grant 326532-2011, the research of M. Coons was supported by ARC Grant DE140100223, and the research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia

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