Factors with prescribed number of invariant subalgebras not arising from subgroups

TL;DR

This paper constructs specific groups with exactly n invariant von Neumann subalgebras that do not originate from subgroups, for any given integer n.

Contribution

It provides explicit constructions of groups with a prescribed number of invariant subalgebras not associated with subgroups.

Findings

For each n, constructed i.c.c. groups G with exactly n such invariant subalgebras.

The associated II$_1$ factors L(G) have precisely n G-invariant von Neumann subalgebras.

These subalgebras do not come from subgroup structures.

Abstract

For any given integer $n\geq 1$, we construct i.c.c. groups $G$ such that the II$_1$ factors $L(G)$ have exactly $n$-many $G$-invariant von Neumann subalgebras not arising from subgroups.