A class of finitely generated groups with irrational translation numbers

Abstract. In this paper we describe a class of examples of groups having word metrics with irrational translation numbers, namely groups of the form <FORMULA>&gif1;</FORMULA>, n ≥3 where <FORMULA>$ {\mit\phi} \in {\rm \ GL} (n,{\Bbb Z}) $</FORMULA> is irreducible and has precisely two eigenvalues of unit modulus. These translation numbers can be computed explicitly. In addition, we prove that if the group G is of the form <FORMULA>&gif2;</FORMULA> where <FORMULA>$ {\mit\gamma} \in {\rm \ GL} (n,{\Bbb Z}) $</FORMULA>, n ≥2 is irreducible and has an eigenvalue of unit modulus then G has a (indiscrete) cocompact faithful action by translations on <FORMULA>$ {\Bbb R}^3 $</FORMULA>.