About the obstacle to proving the Baum–Connes conjecture without coefficient for a non-cocompact lattice in $S{p}_4$ in a local field

Abstract

We introduce property $(T_{\mathrm{Schur}},G,K)$ and prove it for some non-cocompact lattice in $S{p}_4$ 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.