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Abstract. In this paper we describe a class of examples of groups having word metrics with irrational translation numbers, namely groups of the form &gif1;, 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 &gif2; 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 $.