Abstract
Some rigidity results for II$_1$ factors arising from wreath products of property (T) groups
Ionut Chifan
University of Iowa
We show that any infinite collection \( (\Gamma_n)_{n \in \mathbb{N}} \) of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon.
If \( \Lambda \) is an arbitrary group such that \( L(\bigoplus_{n \in \mathbb{N}} \Gamma_n) \cong L(\Lambda) \), then there exists an infinite direct sum decomposition
\[
\Lambda = \left( \bigoplus_{n \in \mathbb{N}} \Lambda_n \right) \oplus A,
\]
with \( A \) icc amenable such that, for all \( n \in \mathbb{N} \), up to amplifications, we have
\[
L(\Gamma_n) \cong L(\Lambda_n)
\quad \text{and} \quad
L\!\left(\bigoplus_{k \geq n} \Gamma_k \right) \cong L\!\left( \left(\bigoplus_{k \geq n} \Lambda_k \right) \oplus A \right).
\]
The result is sharp and complements the previous finite product rigidity property found in the literature. Using this, we provide an uncountable family of restricted wreath products
\[
\Gamma = \Sigma \wr \Delta
\]
of icc, property (T) groups \( \Sigma \), \( \Delta \), whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras \( L(\Gamma) \).
Along the way we highlight several applications of these results to the study of rigidity in the \( \mathbb{C}^* \)-algebra setting. This is based on joint work with Bogdan Udrea.
If \( \Lambda \) is an arbitrary group such that \( L(\bigoplus_{n \in \mathbb{N}} \Gamma_n) \cong L(\Lambda) \), then there exists an infinite direct sum decomposition
\[
\Lambda = \left( \bigoplus_{n \in \mathbb{N}} \Lambda_n \right) \oplus A,
\]
with \( A \) icc amenable such that, for all \( n \in \mathbb{N} \), up to amplifications, we have
\[
L(\Gamma_n) \cong L(\Lambda_n)
\quad \text{and} \quad
L\!\left(\bigoplus_{k \geq n} \Gamma_k \right) \cong L\!\left( \left(\bigoplus_{k \geq n} \Lambda_k \right) \oplus A \right).
\]
The result is sharp and complements the previous finite product rigidity property found in the literature. Using this, we provide an uncountable family of restricted wreath products
\[
\Gamma = \Sigma \wr \Delta
\]
of icc, property (T) groups \( \Sigma \), \( \Delta \), whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras \( L(\Gamma) \).
Along the way we highlight several applications of these results to the study of rigidity in the \( \mathbb{C}^* \)-algebra setting. This is based on joint work with Bogdan Udrea.