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$\mathrm{SL}_{4}(\mathbf{Z})$ is not purely matricial field

Michael Magee, Mikael de la Salle

TL;DR

The paper proves that every finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbb{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbb{Z})$-invariant vector, showing that $\mathrm{SL}_{4}(\mathbb{Z})$ is not purely matricial field. The authors reduce the problem to prime powers using the congruence subgroup property and CRT, then perform a three-step analysis on $\mathrm{SL}_{4}(\mathbb{Z}/p^{r}\mathbb{Z})$ representations, leveraging unipotent and Heisenberg-type subgroups, dual actions, and induced representations to extract an $\mathrm{SL}_{2}$-invariant vector. A key component is establishing that certain elementary subgroups act non-trivially and then applying Frobenius reciprocity to deduce invariants. The work clarifies the MF landscape for $\mathrm{SL}_{4}(\mathbb{Z})$, notes a Deligne-provided counterexample in the $\mathrm{SL}_{3}$ case, and discusses implications for operator-norm behavior and the relationship between property (T) and MF-ness, contributing to the broader understanding of MF reduced C*-algebras of arithmetic groups.

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 $C^*$-algebra into an ultraproduct of matrix algebras.

$\mathrm{SL}_{4}(\mathbf{Z})$ is not purely matricial field

TL;DR

The paper proves that every finite dimensional unitary representation of contains a non-zero -invariant vector, showing that is not purely matricial field. The authors reduce the problem to prime powers using the congruence subgroup property and CRT, then perform a three-step analysis on representations, leveraging unipotent and Heisenberg-type subgroups, dual actions, and induced representations to extract an -invariant vector. A key component is establishing that certain elementary subgroups act non-trivially and then applying Frobenius reciprocity to deduce invariants. The work clarifies the MF landscape for , notes a Deligne-provided counterexample in the case, and discusses implications for operator-norm behavior and the relationship between property (T) and MF-ness, contributing to the broader understanding of MF reduced C*-algebras of arithmetic groups.

Abstract

We prove that every finite dimensional unitary representation of contains a non-zero -invariant vector. As a consequence, there is no sequence of finite-dimensional representations of that gives rise to an embedding of its reduced -algebra into an ultraproduct of matrix algebras.
Paper Structure (7 sections, 3 theorems, 34 equations)

This paper contains 7 sections, 3 theorems, 34 equations.

Table of Contents

  1. Statement of results
  2. Proofs of results
  3. Reduction to prime powers
  4. Prime powers: step 1
  5. Prime powers: step 2
  6. Prime powers: step 3
  7. Representations of $\mathrm{SL}_3(\mathbf{Z}/p\mathbf{Z})$

Key Result

Theorem 1.1

Every finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector.

Theorems & Definitions (6)

  • Theorem 1.1
  • Definition 1.2
  • Corollary 1.3
  • proof : Proof of Corollary \ref{['cor:-is-not']}
  • Proposition 2.1
  • Example 2.2