About the obstacle to proving the Baum–Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field

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We introduce property (T_\mathrm {Schur},G,K) and prove it for some non-cocompact lattice in Sp_4 in a local field of finite characteristic. We show that property (T_\mathrm {Schur},G,K) for a non-cocompact lattice \Gamma in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for \Gamma with known methods, and this is stronger than the well-known fact that \Gamma does not have the property of rapid decay (property (RD)). It is the first example (as announced in [7]) for which all known methods for proving the Baum–Connes conjecture without coefficient fail.