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Twisted $L^2$-Betti numbers of sofic groups

Jan Boschheidgen, Andrei Jaikin-Zapirain

Abstract

Wolfang Lück asked if twisted $L^2$-Betti numbers of a group are equal to the usual $L^2$-Betti numbers rescaled by the dimension of the twisting representation. We confirm this for sofic groups.

Twisted $L^2$-Betti numbers of sofic groups

Abstract

Wolfang Lück asked if twisted -Betti numbers of a group are equal to the usual -Betti numbers rescaled by the dimension of the twisting representation. We confirm this for sofic groups.
Paper Structure (8 sections, 4 theorems, 42 equations)

This paper contains 8 sections, 4 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Von Neumann Sylvester matrix rank function
  5. Sofic groups and the sofic Lück approximation
  6. The proof of the main result
  7. The algebraic case
  8. Proof of Theorem \ref{['twistedsofic']}

Key Result

Theorem 1.1

The Lück twisted conjecture holds for sofic groups.

Theorems & Definitions (19)

  • Conjecture 1: The independence conjecture
  • Conjecture 2: The Lück twisted conjecture
  • Theorem 1.1
  • Theorem 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['algebraic']}
  • Claim 3.3
  • proof
  • ...and 9 more