Interplay of structure and magnetism in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>LuFe</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msub><mml:mi>Ge</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> tuned by hydrostatic pressure

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.