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Archiv der Mathematik

, Volume 69, Issue 4, pp 265–274 | Cite as

A class of finitely generated groups with irrational translation numbers

  • Gregory R. Conner

Abstract.

In this paper we describe a class of examples of groups having word metrics with irrational translation numbers, namely groups of the form ℤ n ×φ, n ≥3 where \( {\mit\phi} \in {\rm \ GL} (n,{\Bbb Z}) \) is irreducible and has precisely two eigenvalues of unit modulus. These translation numbers can be computed explicitly. In addition, we prove that if the group G is of the form G=ℤ n ×γ where \( {\mit\gamma} \in {\rm \ GL} (n,{\Bbb Z}) \), n ≥2 is irreducible and has an eigenvalue of unit modulus then G has a (indiscrete) cocompact faithful action by translations on \( {\Bbb R}^3 \).

Keywords

Faithful Action Unit Modulus Translation Number Word Metrics Irrational Translation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 1997

Authors and Affiliations

  • Gregory R. Conner
    • 1
  1. 1.Mathematics Department, Brigham Young University, Provo, UT 84602, USAUS

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