A group is conjugacy separable if the conjugacy class of every element is closed in the profinite topology. We prove that free-by-cyclic groups with unipotent and polynomially growing monodromy are conjugacy separable. Along the way, we show that double cosets of cyclic subgroups are separable and cyclic subgroups satisfy the Wilton--Zalesskii property. Our methods involve constructing vertex fillings and p-quotients.
This is based on joint work with Francois Dahmani, Sam Hughes and Nicholas Touikan.
