we show that a graph product of tracial von neumann algebras is strongly $1$-bounded if the first $\ell^2$-betti number vanishes for an associated dense $*$-subalgebra.  graph products of tracial von neumann algebras were studied by caspers and fima, and generalize green's graph product of groups.  given groups $g_v$ for each vertex of a graph $\gamma$, the graph product is the free product modulo the relations that $g_v$ and $g_w$ commute when $v \sim w$; for von neumann algebras, graph products are described by a certain moment relation.  in our paper, the crux of the argument is a generalization to tracial von neumann algebras of the statement that soficity of groups is preserved by graph products.  we replace soficity for groups with a more general notion of algebraic soficity for a $*$-algebra $a$, which means the existence of certain approximations for the generators of $a$ by matrices with algebraic integer entries and approximately constant diagonal.  we show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of charlesworth and collins as well as au-c{\'e}bron-dahlqvist-gabriel-male.  in particular, this gives a new probabilistic proof of ciobanu-holt-rees's result that soficity of groups is preserved by graph products.

this is based on joint work with rolando de santiago, ben hayes, srivatsav kunnawalkam elayavalli, brent nelson.