Internal graphs of graph products of hyperfinite ${\mathrm{II}}_1$-factors

Abstract

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph $\mathrm{Int}(\Gamma )$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite ${\mathrm{II}}_1$-factors $R_{\Gamma }$. In particular, we can classify $R_{\Gamma }$ for some typical types of graphs, including lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite ${\mathrm{II}}_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than $1$. Our proof is based on the recent resolution of the Peterson–Thom conjecture.