Relative solidity results and their applications to computations of some II$_1$ factor invariants

Abstract: In this paper we prove that whenever $G$ is hyperbolic relative to a family of exact, ressidually finite subgroups ${H_1, \ldots, H_n}$, the corresponding von Neumann algebra $\mathcal L(G)$ is solid relative to the family of subalgebras ${\mathcal L(H_1),\ldots ,\mathcal L(H_n)}$. Building on this result and combining it with findings from geometric group theory, we construct a continuum of icc property (T) relative hyperbolic groups that give rise to pairwise non virtually isomorphic factors, each of which has trivial one-sided fundamental semigroup.