Strong 1-boundedness, L2-Betti numbers, algebraic soficity, and graph products

We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying * -algebra has vanishing first L 2 -Betti number.The proof uses a combination of the following two key ideas to obtain lower bounds on the Fuglede-Kadison determinant of matrix polynomials in a generating set: a notion called "algebraic soficity" for * -algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a probabilistic construction of the authors of permutation models for graph independence over the diagonal.