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arXiv · 2505.05179

Internal graphs of graph products of hyperfinite II$_1$-factors

Martijn Caspers·Enli Chen
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Abstract

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph ${\rm Int}(\Gamma)$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_{\Gamma}$. In particular, we can classify $R_{\Gamma}$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite 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.

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Martijn Caspers, Enli Chen. 2025-05-08. Internal graphs of graph products of hyperfinite II$_1$-factors. https://arxiv.org/abs/2505.05179

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