Approximation properties of fixed point planar algebras

Let $(\Gamma,\mu)$ be a bipartite graph together with a weight on its vertices. Assume that $\mu$ is an eigenvector for the adjacency matrix of $\Gamma$. Let Aut$(\Gamma, \mu)$ be the automorphism group of the bipartite graph $\Gamma$ that scales the weight $\mu$. It is a locally compact totally disconnected group that acts on the bipartite graph planar algebra $P$ associated to $(\Gamma,\mu)$. Consider a subgroup G < Aut$(\Gamma, \mu)$ and the set of fixed points $P^G \subset P$ that we assume to be a subfactor planar algebra. If the closure of G inside Aut$(\Gamma, \mu)$ satisfies an approximation property such as amenability, the Haagerup property, weak amenability, or not having property (T), then the subfactor planar algebra $P^G$ inherits this property respectively. As a corollary we show that if $\Gamma$ is a tree, then the subfactor planar algebra $P^G$ has the Haagerup property and has the complete metric approximation property (CMAP). This provides an infinite family of subfactor planar algebras that have non-integer index, are non-amenable, have the Haagerup property, and have CMAP. We define the crossed product of a (finite) von Neumann algebra by a Hecke pair of groups. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras are described by such a crossed product including Bisch-Haagerup subfactors.