Free products of sofic groups with amalgamation over monotileably amenable groups

We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that families of independent uniformly distributed permutation matrices and certain families of non-random permutation matrices (essentially, those coming from quasi--actions of a sofic group) are asymptotically *-free as the matrix size grows without bound.