A Unique Prime Decomposition Result for Wreath Product Factors

TL;DR

This paper proves unique tensor product decomposition results for certain II$_1$ factors using advanced deformation and rigidity techniques, also extending to measure equivalence of related groups.

Contribution

It introduces new uniqueness results for wreath product factors in the framework of Popa's deformation/rigidity theory.

Findings

Unique tensor product decomposition for wreath product factors.

Extension of results to measure equivalence of group products.

Application of malleable deformations and spectral gap rigidity.

Abstract

We use malleable deformations combined with spectral gap rigidity theory, in the framework of Popa's deformation/rigidity theory to prove unique tensor product decomposition results for II$_1$ factors arising as tensor product of wreath product factors. We also obtain a similar result regarding measure equivalence decomposition of direct products of such groups.