Relative solidity results for von Neumann algebras and their applications to W*-rigidity questions and invariant computations

Von Neumann algebras were first introduced by John von Neumann in 1926 as a mathematical tool to model particle physics. Then, in 1943, he, together with Francis Murray, found a natural way to associate von Neumann algebras with countable discrete groups. Understanding the structure of von Neumann algebras is a key area of research in operator algebra. A fundamental question in this field is how various properties of groups are reflected in the corresponding von Neumann algebras. A number of mathematicians have discovered examples demonstrating that this is not always true. For instance, while lamplighter groups cannot be non-trivially decomposed as products, their corresponding von Neumann algebra can be expressed in various ways as nontrivial tensor products. In other cases, there have been examples of groups such as Relative hyperbolic groups where the commuting parts in their von Neumann algebra can be identified using the commuting parts of the corresponding groups using deformation/rigidity theory. A key aspect of this structural theory involves understanding commuting von Neumann subalgebras and their rigidity properties, as it helps distinguish a system’s classical and quantum components.

In this thesis, we investigate commuting subalgebras of von Neumann algebras associated with relatively hyperbolic groups. In particular, we make new progress on a relative solidity conjecture inspired by Ozawa’s solidity theorem, which has been a subject of interest in the study of rigidity phenomena in operator algebras. Using our relative solidity results, we establish several W*-rigidity theorems, including the construction of a continuum of pairwise non-virtually isomorphic icc property (T) relatively hyperbolic groups whose von Neumann algebras exhibit unique structural properties. Notably, each of these von Neumann algebras has a trivial one-sided fundamental semigroup. These results contribute to ongoing developments in deformation/ rigidity theory and provide new insights into Connes’ Rigidity Conjecture and related problems.