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We show vanishing of the second $L^p$-cohomology group for most semisimplealgebraic groups of rank at least 3 over local fields. More precisely, we showthis result for $\SL(4)$, for simple groups of rank $\geq 4$ that are not ofexceptional type or of type $D_4$ and for all semisimple, non-simple groups ofrank $\geq 3$. Our methods work for large values of $p$ in the real case andfor all $p>1$ in the non-Archimedean case. This result points towards apositive answer to Gromov's question on vanishing of $L^p$-cohomology ofsemisimple groups for all $p>1$ in degrees below the rank. The methods consistin using a spectral sequence \`a la Bourdon-R\'emy, adapting a version ofMautner's phenomenon from Cornulier-Tessera and concluding thanks to acombinatorial case-by-case study of classical simple groups.

Vanishing of the second $L^p$-cohomology group for most semisimple groups of rank at least 3