The planar algebra of group-type subfactors

TL;DR

This paper computes the planar algebra of group-type subfactors arising from group actions on II$_1$ factors and characterizes all subfactors with such planar algebras as originating from this construction.

Contribution

It explicitly determines the planar algebra of group-type subfactors and proves a classification result linking these planar algebras to the subfactors constructed from group actions.

Findings

Computed the planar algebra for group-type subfactors.

Proved that all subfactors with a group-type planar algebra come from the described construction.

Explicitly described the action of Jones' planar operad on these subfactors.

Abstract

If $G$ is a countable, discrete group generated by two finite subgroups $H$ and $K$ and $P$ is a II$_1$ factor with an outer G-action, one can construct the group-type subfactor $P^H \subset P \rtimes K$ introduced in \cite{BH}. This construction was used in \cite{BH} to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra (in the sense of Jones \cite{J2}) of this subfactor and prove that any subfactor with an abstract planar algebra of "group type" arises from such a subfactor. The action of Jones' planar operad is determined explicitly.