Toward the "perfect square" ring laser gyroscope

He-Ne ring lasers are, actually, the most sensitive devices for inertial rotation measurements. Depending on their linear dimensions, they find applications in different fields: from inertial navigation to structural engineering, from metrology to geophysics and fundamental physics. Here we report the recent progresses toward the development of GINGER- Gyroscopes IN GEneral Relativity, a triaxial ultra-sensitive ring laser gyroscope whose primary target is a ground measurement the Lense-Thirring effect, a small shift of the Earth rotation rate foreseen by Einstein's General Relativity. This target implies a measurement of the Earth rotation rate with a sensitivity better than one part over 1010 and so requires a high level of stability and accuracy of the laser array and of the gyroscope stability. The main part of the research activity presented in this paper involves, then, a strict control of the systematic errors related to the fluctuation of the optical cavity geometry and of the laser parameters.