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

In this paper we prove that whenever $G$ is hyperbolic relative to a family of exact, ressidually finite subgroups $\{H_1,\dots ,H_n\}$, the corresponding von Neumann algebra $\mathcal{L}(G)$ is solid relative to the family of subalgebras $\{\mathcal{L}(H_1),\dots ,\mathcal{L}(H_n)\}$. 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.