We consider the model of an uncharged massive particle linearly coupled to a quantized radiation field, widely referred to as Nelson model. Despite having the character of a toy model from the physics perspective it acts as an important instance for the mathematical treatment of quantum field theories, since it exhibits both ultraviolet and infrared divergences. Nelson (1964) orginally proved that the ultraviolet problem can be treated by self-energy renormalization. Explicitly, adding an appropriately chosen counter term to the Hamilton operator with an ultraviolet cutoff, the operator converges to the renormalized Nelson Hamiltonian in the norm resolvent sense. Due to the translation-invariance of the model, both the Hamiltonian with cutoff and the renormalized Hamiltonian decompose into direct integrals of fiber operators describing the system at fixed total momentum. In this talk, we discuss a necessary and sufficient condition for the domain of the renormalized fiber operators to be independent of the total momentum. Then, we sketch our recent proof for the absence of ground states of the renormalized fiber operators, which is a consequence of the infrared divergence of the model. Along the way, we will relate this work to further recent results on the renormalized Nelson model. Based on joint work with Thomas Norman Dam.