  LuFe$_4$Ge$_2$ crystallizes in the ZrFe$_4$Si$_2$-type structure, hosting
chains of Fe-tetrahedra giving rise to geometric frustration and
low-dimensionality. The compound orders antiferromagnetically at around 36 K
accompanied by a simultaneous structural transition from a tetragonal to an
orthorhombic phase. The hydrostatic pressure dependence of the magnetic and
structural transitions is investigated using electrical-transport, ac
magnetic-susceptibility, ac calorimetry, M$\ddot{\rm o}$ssbauer, muon-spin
relaxation ($\mu$SR), and x-ray diffraction measurements. External pressure
suppresses the first-order transition to the antiferromagnetic phase (AFM1)
around 1.8 GPa. The structural transition is largely unaffected by pressure and
remains between 30 to 35 K for pressures up to 2 GPa. A second
antiferromagnetic phase (AFM2) is observed at higher pressures. The transition
from the paramagnetic to the AFM2 phase is of second-order nature and appears
to be connected to the structural transition. The magnetic volume fraction
obtained from $\mu$SR and M$\ddot{\rm o}$ssbauer measurements reveal that the
entire sample undergoes magnetic ordering in both magnetic phases. In addition,
similar low-temperature muon-precession frequencies in AFM1 and AFM2 phases
point at similar ordered moments and magnetic structures in both phases. Our
results further indicate enhanced magnetic fluctuations in the pressure induced
AFM2 phase. The experimental observations together with density functional
theory calculations suggest that the magnetic and structural order parameters
in LuFe$_4$Ge$_2$ are linked by magnetic frustration, causing the simultaneous
magneto-structural transition.
