Tensor product decompositions of II$_1$ factors arising from extensions of amalgamated free product groups

TL;DR

This paper introduces a new family of ICC groups with a product rigidity property, ensuring all tensor product decompositions of their associated II$_1$ factors correspond only to the group's direct product decompositions, with applications to prime factor examples.

Contribution

The paper constructs a new class of ICC groups from HNN-extensions and amalgamated free products exhibiting tensor product rigidity in their II$_1$ factors, including many notable groups.

Findings

All tensor product decompositions of $L(Gamma)$ derive from group decompositions.

Many constructed groups produce prime II$_1$ factors.

Includes groups like graph products, Higman, and Cremona groups.

Abstract

In this paper we introduce a new family of icc groups $\Gamma$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(\Gamma)$ arise only from the canonical direct product decompositions of the underlying group $\Gamma$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence, we obtain several new examples of groups that give rise to prime factors.