Expanding maps on infra-nilmanifolds of homogeneous type

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient $E\backslash L$, where $L$ is a connected and simply connected nilpotent Lie group and $E$is a torsion-free uniform discrete subgroup of $L {\mathbb o} C$, with $C$ a compact subgroup of $\operatorname{Aut}(L)$. We show that if the Lie algebra of $L$ is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.