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On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group

Marcos Escartín Ferrer

TL;DR

This paper determines the first $\,\ell^2$-Betti numbers of the automorphism and outer automorphism groups of right-angled Artin groups (RAAGs), giving sharp criteria for when these invariants are nonzero and linking them to algebraic fibring. It develops a detailed analysis of the pure symmetric automorphism groups and their outer counterparts, uses BNS invariants to fully classify algebraic fibering in the transvection-free setting, and characterizes when Out$(A_ abla)$ virtually fibers. The work also provides comprehensive descriptions of the transvection subgroup and its impact on $eta^{(2)}$-numbers, supplemented by explicit RAAG examples illustrating when higher $\,\ell^2$-Betti numbers vanish or persist and when virtual fibrations occur. Collectively, these results illuminate the interplay between analytic invariants and the fibering structure of RAAG automorphism groups, with concrete constructions showing a wide range of fibring phenomena and higher-dimensional Betti behavior.

Abstract

We compute the first $\ell^2$-Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups $\mathrm{PSA}(A_Γ)$ and $\mathrm{PSO}(A_Γ)$ and the virtual algebraic fibring of $\mathrm{Out}(A_Γ)$ in the case when $A_Γ$ admits no non-inner partial conjugation. In the transvection-free case, we show that $β_1^{(2)}(\mathrm{Out}(A_Γ)) = 0$ if and only if $\mathrm{Out}(A_Γ)$ virtually fibres.

On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group

TL;DR

This paper determines the first -Betti numbers of the automorphism and outer automorphism groups of right-angled Artin groups (RAAGs), giving sharp criteria for when these invariants are nonzero and linking them to algebraic fibring. It develops a detailed analysis of the pure symmetric automorphism groups and their outer counterparts, uses BNS invariants to fully classify algebraic fibering in the transvection-free setting, and characterizes when Out virtually fibers. The work also provides comprehensive descriptions of the transvection subgroup and its impact on -numbers, supplemented by explicit RAAG examples illustrating when higher -Betti numbers vanish or persist and when virtual fibrations occur. Collectively, these results illuminate the interplay between analytic invariants and the fibering structure of RAAG automorphism groups, with concrete constructions showing a wide range of fibring phenomena and higher-dimensional Betti behavior.

Abstract

We compute the first -Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups and and the virtual algebraic fibring of in the case when admits no non-inner partial conjugation. In the transvection-free case, we show that if and only if virtually fibres.

Paper Structure

This paper contains 19 sections, 46 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\ell^2$-Betti numbers
  4. Algebraic fibring
  5. Right-angled Artin groups
  6. Automorphism group of a RAAG
  7. Pure symmetric automorphism group
  8. Algebraic fibring
  9. Pure symmetric automorphism group
  10. Transvection subgroup of $\mathop{\mathrm{Out}}\nolimits(A_\Gamma)$
  11. $\ell^2$-Betti numbers
  12. Transvection subgroup of $\mathop{\mathrm{Out}}\nolimits(A_\Gamma)$
  13. First $\ell^2$-Betti number of $\mathop{\mathrm{Aut}}\nolimits(A_\Gamma)$ and $\mathop{\mathrm{Out}}\nolimits(A_\Gamma)$
  14. Higher dimensional $\ell^2$-Betti numbers
  15. Examples
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $A_\Gamma$ be a RAAG. Then, $\beta_1^{(2)}(\mathop{\mathrm{Aut}}\nolimits(A_\Gamma)) > 0$ if and only if $A_\Gamma \cong \mathbb{Z}^2$ and $\beta_1^{(2)}(\mathop{\mathrm{Out}}\nolimits(A_\Gamma)) > 0$ if and only if one of the following conditions holds:

Theorems & Definitions (91)

  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: Kammeyer, Exercise 4.4.1
  • Proposition 2.2: Kammeyer, Theorem 4.15 (iii)
  • Proposition 2.3
  • proof
  • Theorem 2.4: Kammeyer, Theorem 4.15 (i)
  • ...and 81 more