On groups with Property [formula omitted

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Abstract Let 1 p ∞ , p ≠ 2 . Property ( T l p ) for a second countable locally compact group G is a weak version of Kazhdan Property (T), defined in terms of the orthogonal representations of G on l p . Property ( T l p ) is characterized by an isolation property of the trivial representation from the monomial unitary representations of G associated to open subgroups. Connected groups with Property ( T l p ) are the connected groups with a compact abelianization. In the case of a totally disconnected group, isolation of the trivial representation from the quasi-regular representations associated to open subgroups suffices to characterize Property ( T l p ) . Groups with Property ( T l p ) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of k-regular trees for k ≥ 3 have Property ( T l p ) . In the case of discrete groups, Property ( T l p ) implies Lubotzky's Property (τ) and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice Γ in a product G 1 × G 2 of locally compact groups has Property ( T l p ) , whenever G 1 has Property (T) and G 2 is connected and minimally almost periodic. Such a lattice does not have Property (T) if G 2 does not have Property (T).