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Property (T) group factors whose Jones index set equals all positive integers

Ionut Chifan, Junhwi Lim

TL;DR

The paper addresses the Jones index problem for property (T) II$_1$ factors by constructing a continuum of pairwise non-stably isomorphic factors $\mathcal{L}(G)$ arising from generalized wreath-like product groups $G\in\mathcal{WR}(A,B\curvearrowright I)$ with abelian base. It develops a rigidity framework for virtual $\ast$-isomorphisms between such factors, showing any $\Theta:\mathcal{L}(G)\to\mathcal{L}(H)^t$ enforces an integral amplification $t\in\mathbb{N}$ and reduces to a finite collection of finite-index subgroups $K_i\le G$, injective maps $\gamma_i:K_i\to H$, and induced representations $\mathrm{Ind}^G_{K_i}(\pi_{\gamma_i,\rho_i})$, yielding a precise diagonal form for $\Theta(u_g)$. This machinery yields an index formula $[\mathcal{L}(H)^t: \Theta(\mathcal{L}(G))]= t\sum_i s_i[H:\gamma_i(K_i)]$, enabling a control over possible Jones indices. Consequently, the authors show $\mathscr{I}(\mathcal{L}(G))\subseteq\mathbb{N}$ and construct a continuum of ICC property (T) wreath-like products with $\mathscr{I}(\mathcal{L}(G_j))=\mathbb{N}$; every positive integer is realized as the index of an irreducible finite-index subfactor via finite-index extensions. This yields a broad class of property (T) group factors with full integer Jones index sets and demonstrates the strength of Connes rigidity phenomena in this setting, advancing open questions about Jones indices for ICC property (T) groups.

Abstract

Using a mélange of techniques at the rich intersection of deformation/rigidity theory, finite index subfactor theory, and geometric group theory, we prove the existence of a continuum of property (T) factors that are pairwise non-stably isomorphic and whose Jones index sets consist of all positive integers. These factors are realized as group von Neumann algebras $\mathcal{L}(G)$ associated with property (T) generalized wreath-like product groups $G \in \mathscr{WR}(A, B \curvearrowright I)$ introduced in [CIOS23b], where $A$ is abelian, $B$ is a non-parabolic subgroup of a relatively hyperbolic group with residually finite peripheral structure, and $B \curvearrowright I$ is a faithful action with infinite orbits. Integer index subfactors of $\mathcal{L}(G)$ are constructed from extensions of $G$. This result advances an open question of P. de la Harpe [dlH95].

Property (T) group factors whose Jones index set equals all positive integers

TL;DR

The paper addresses the Jones index problem for property (T) II factors by constructing a continuum of pairwise non-stably isomorphic factors arising from generalized wreath-like product groups with abelian base. It develops a rigidity framework for virtual -isomorphisms between such factors, showing any enforces an integral amplification and reduces to a finite collection of finite-index subgroups , injective maps , and induced representations , yielding a precise diagonal form for . This machinery yields an index formula , enabling a control over possible Jones indices. Consequently, the authors show and construct a continuum of ICC property (T) wreath-like products with ; every positive integer is realized as the index of an irreducible finite-index subfactor via finite-index extensions. This yields a broad class of property (T) group factors with full integer Jones index sets and demonstrates the strength of Connes rigidity phenomena in this setting, advancing open questions about Jones indices for ICC property (T) groups.

Abstract

Using a mélange of techniques at the rich intersection of deformation/rigidity theory, finite index subfactor theory, and geometric group theory, we prove the existence of a continuum of property (T) factors that are pairwise non-stably isomorphic and whose Jones index sets consist of all positive integers. These factors are realized as group von Neumann algebras associated with property (T) generalized wreath-like product groups introduced in [CIOS23b], where is abelian, is a non-parabolic subgroup of a relatively hyperbolic group with residually finite peripheral structure, and is a faithful action with infinite orbits. Integer index subfactors of are constructed from extensions of . This result advances an open question of P. de la Harpe [dlH95].

Paper Structure

This paper contains 15 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments.
  4. Preliminaries
  5. Cocycle crossed product von Neumann algebras
  6. Finite index inclusions of von Neumann algebras
  7. Construction of finite index subfactors from outer automorphisms
  8. Popa's intertwining techniques
  9. Virtual isomorphisms between II$_1$ factors associated with property (T) wreath-like product groups
  10. Jones index set for II$_1$ factors associated with property (T) wreath-like product groups
  11. The basic construction of ICC group subfactors
  12. Amplificational description of the basic construction
  13. Inclusions of von Neumann algebras of ICC groups
  14. The standard invariant
  15. Cocycle action description

Key Result

Theorem 1

Let $A,C$ be non-trivial abelian groups. Let $B,D$ be nonparabolic ICC subgroups of finitely generated groups which are hyperbolic relative to a finite family of residually finite groups. Let $G\in\mathcal{W}\mathcal{R}(A,B \curvearrowright I)$ and $H\in\mathcal{W}\mathcal{R}(C,D\curvearrowright J)$ Here, $(u_g)_{g\in G}\subset \mathcal{L}(G)$ and $(v_h)_{h\in H}\subset \mathcal{L}(H)$ are the can

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5: Mac63
  • Remark 2.6
  • Theorem 2.7: Sut80B
  • Corollary 2.8
  • ...and 32 more