Vanishing of the second <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -cohomology group for most semisimple groups of rank at least 3

We show vanishing of the second <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -cohomology group for most semisimple algebraic groups of rank at least 3 over local fields. More precisely, we show this result for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , for simple groups of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> that are not of exceptional type or of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> and for all semisimple, non-simple groups of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . Our methods work for large values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> in the real case and for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> in the non-Archimedean case. This result points towards a positive answer to Gromov’s question on vanishing of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -cohomology of semisimple groups for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> in degrees below the rank. The methods consist in using a spectral sequence à la Bourdon–Rémy, adapting a version of Mautner’s phenomenon from Cornulier–Tessera and concluding thanks to a combinatorial case-by-case study of classical simple groups.