Secondary discharges, which consist of the breakdown of a gap near a GEM foil upon a primary discharge across that GEM, are studied in this work. Their main characteristics are the occurrence a few $10,mu textrm{s}$ after the primary, the relatively sharp onset at moderate electric fields across the gap, the absence of increased fields in the system, and their occurrence under both field directions. They can be mitigated using series resistors in the high-voltage connection to the GEM electrode facing towards an anode. The electric field at which the onset of secondary discharges occurs indeed increases with increasing resistance. Discharge propagation form GEM to GEM in a multi-GEM system affects the occurrence probability of secondary discharges in the gaps between neighbouring GEMs. Furthermore, evidence of charges flowing through the gap after the primary discharge are reported. Such currents may or may not lead to a secondary discharge. A characteristic charge, of the order of $10^{10},textrm{electrons}$, has been measured as the threshold for a primary discharge to be followed by a secondary discharge, and this number slightly depends on the gas composition. A mechanism involving the heating of the cathode surface as trigger for secondary discharges is proposed.
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