Relative solidity for biexact groups in measure equivalence

Abstract: We demonstrate a relative solidity property for the product of a nonamenable biexact group with an arbitrary infinite group in the measure equivalence setting. Among other applications, we obtain the following unique product decomposition for products of nonamenable biexact groups, strengthening \cite{Sa09}: for any nonamenable biexact groups $\Gamma_1,\cdots, \Gamma_n$, if a product group $\Lambda_1\times \Lambda_2$ is measure equivalent to $\times_{k=1}^n\Gamma_k$, then there exists a partition $T_1\sqcup T_2={1,\dots, n}$ such that $\Lambda_i$ is measure equivalent to $\times_{k\in T_i}\Gamma_k$ for $i=1,2$.