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


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 \( {\Bbb R}^3 \).


Faithful Action Unit Modulus Translation Number Word Metrics Irrational Translation 
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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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