Link invariants from $L^2$-Burau maps of braids

A previous work of A. Conway and the author introduced $L^2$-Burau maps of braids, which are generalizations of the Burau representation whose coefficients live in a more general group ring than the one of Laurent polynomials. This same work established that the $L^2$-Burau map of a braid at the group of the braid closure yields the $L^2$-Alexander torsion of the braid closure in question, as a variant of the well-known Burau-Alexander formula. In the present paper, we generalize the previous result to $L^2$-Burau maps defined over all quotients of the group of the braid closure. The link invariants we obtain are twisted $L^2$-Alexander torsions of the braid closure, and recover more topological information, such as the hyperbolic volumes of Dehn fillings. The proof needs us to first generalize several fundamental formulas for $L^2$-torsions, which have their own independent interest. We then discuss how likely we are to generalize this process to yet more groups. In particular, a detailed study of the influence of Markov moves on $L^2$-Burau maps and two explicit counter-examples to Markov invariance suggest that twisted $L^2$-Alexander torsions of links are the only link invariants we can hope to build from $L^2$-Burau maps with the present approach.