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Register a new userNon-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups
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We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromovs question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
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We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
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