  We present an extensive study of the ferromagnetic heavy fermion compound
U$_4$Ru$_7$Ge$_6$. Measurements of electrical resistivity, specific heat and
magnetic properties show that U$_4$Ru$_7$Ge$_6$ orders ferromagnetically at
ambient pressure with a Curie temperature $T_{C} = 6.8 \pm 0.3$ K. The low
temperature magnetic behavior of this soft ferromagnet is dominated by the
excitation of gapless spin-wave modes. Our results on the transport properties
of U$_4$Ru$_7$Ge$_6$ under pressures up to $2.49$ GPa suggest that
U$_4$Ru$_7$Ge$_6$ has a putative ferromagnetic quantum critical point (QCP) at
$P_c \approx 1.7 \pm 0.02$ GPa. In the ordered phase, ferromagnetic magnons
scatter the conduction electrons and give rise to a well defined power law
temperature dependence in the resistivity. The coefficient of this term is
related to the spin-wave stiffness and measurements of the very low temperature
resistivity allow to accompany the behavior of this quantity as the the
ferromagnetic QCP is approached. We find that the spin-wave stiffness decreases
with increasing pressure implying that the transition to the non-magnetic Fermi
liquid state is driven by the softening of the magnons. The observed quantum
critical behavior of the magnetic stiffness is consistent with the influence of
disorder in our system. At quantum criticality ($P = P_c \approx 1.7 \pm 0.02$
GPa), the resistivity shows the behavior expected for an itinerant metallic
system near a ferromagnetic QCP.
