About the difficulty to prove the Baum Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field

TL;DR

This paper introduces a new property related to Schur multipliers and demonstrates its presence in certain lattices, showing it obstructs proving the Baum Connes conjecture without coefficients for these groups, highlighting limitations of current methods.

Contribution

The paper defines property $(T_{Schur},G,K)$ and proves it for specific non-cocompact lattices in $Sp_4$, establishing a new obstacle to the Baum Connes conjecture without coefficients.

Findings

Property $(T_{Schur},G,K)$ holds for some non-cocompact lattices in $Sp_4$.

This property obstructs proving the Baum Connes conjecture without coefficients using known methods.

It provides the first example where all existing methods fail to prove the conjecture for these groups.

Abstract

We introduce property $(T_{Schur},G,K)$ and prove it for some non-cocompact lattice in $Sp_4$ in a local field of finite characteristic. We show that property $(T_{Schur},G,K)$ 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 [Laf10a]) for which all known methods for proving the Baum Connes conjecture without coefficient fail.