  Using an interplay between geometric methods in group theory and soft von
Neuman algebraic techniques we prove that for any icc, acylindrically
hyperbolic group $\Gamma$ its von Neumann algebra $L(\Gamma)$ satisfies the
so-called ISR property: \emph{any von Neumann subalgebra $N\subseteq L(\Gamma)$
that is normalized by all group elements in $\Gamma$ is of the form $N=
L(\Sigma)$ for a normal subgroup $\Sigma \lhd \Gamma$.} In particular, this
applies to all groups $\Gamma$ in each of the following classes: all icc
(relatively) hyperbolic groups, most mapping class groups of surfaces, all
outer automorphisms of free groups with at least three generators, most graph
product groups arising from simple graphs without visual splitting, etc. This
result answers positively an open question of Amrutam and Jiang from
\cite{AJ22}.
  In the second part of the paper we obtain similar results for factors
associated with groups that admit nontrivial (quasi)cohomology valued into
various natural representations. In particular, we establish the ISR property
for all icc, nonamenable groups that have positive first $L^2$-Betti number and
contain an infinite amenable subgroup.
