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

TL;DR

This paper constructs a continuum of property (T) factors with all positive integers as their Jones index set, using advanced techniques in deformation, rigidity, subfactor theory, and geometric group theory.

Contribution

It introduces new property (T) factors with all positive integer Jones indices, realized via group von Neumann algebras of generalized wreath-like product groups.

Findings

Existence of uncountably many non-stably isomorphic property (T) factors

Construction of factors with all positive integer Jones indices

Advancement of open question by P. de la Harpe

Abstract

Using a m\'elange 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].