On groups with Property (T_(?_p))

摘要

Let 1 < p < ∞,p ≠ 2. Property (T_(?_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 ?_p. Property (T_(?_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_(?_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_(?_p)). Groups with Property (T_(?_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 κ-regular trees for κ ≥ 3 have Property (T_(?_p)). In the case of discrete groups, Property (T_(?_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_(?_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).