We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explic...

Thouless pumping is a fundamental instance of quantized transport, which topologically protected. Although its theoretical importance, the adiabaticity condition an obstacle for further practical applications. Here, focusing on Rice-Mele model, we provide family finite-frequency examples that ensure both absence excitations and perfect quantization pumped charge at end each cycle. This family, ...