Explicit computations of all finite index bimodules for a family of II_1 factors

TL;DR

This paper explicitly computes all finite index bimodules for certain II_1 factors arising from generalized Bernoulli actions, revealing their structure and fusion algebra, and providing new examples with trivial fusion algebra.

Contribution

It provides the first explicit computation of the fusion algebra for a class of II_1 factors and characterizes finite index bimodules via group and action commensurability.

Findings

Finite index bimodules correspond to group and action commensurability.

Fusion algebra identified with an extended Hecke algebra.

Constructs examples of II_1 factors with trivial fusion algebra.

Abstract

We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in particular, every isomorphism between M and N) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M-M-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II_1 factor. We obtain in particular explicit examples of II_1 factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.