${\mathrm{\ell }}^2$-Betti numbers and $\mathrm{\Sigma }$-invariants of graph products of cyclic groups and their automorphisms

We study the ${\mathrm{\ell }}^2$-Betti numbers of graph products of cyclic groups. We relate these invariants to the ${\mathrm{\ell }}^2$-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first ${\mathrm{\ell }}^2$-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first ${\mathrm{\ell }}^2$-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.