${\mathrm{SL}}_4(\mathbf{Z})$ is not purely matricial field

Abstract

We prove that every finite dimensional unitary representation of ${\mathrm{SL}}_4(\mathbf{Z})$ contains a non-zero ${\mathrm{SL}}_2(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of ${\mathrm{SL}}_4(\mathbf{Z})$ that gives rise to an embedding of its reduced $C^{\ast }$-algebra into an ultraproduct of matrix algebras.

v2: 9 pages; details added in proof, and example for SL3 made explicit. To appear in Comptes Rendus Mathématique

References in corpus (2)

• Strongly convergent unitary representations of limit groups
• Finite dimensional approximations of certain amalgamated free products of groups