Residual finiteness properties of some of Halls groups

In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In particular we obtain a group that is conjugacy separable with solvable word problem but unsolvable conjugacy problem, we obtain a group with large conjugacy separability growth but small conjugator length function and residual finiteness growth, and we also obtain both a class of groups that for most functions $f:\mathbb{N}\rightarrow\mathbb{N}$ larger then $n^3$, contain a group $G$ such that the conjugator length of $G$ is given by $f$, as well as a group where the conjugator length is superlinear but subquadratic. On the other hand we also consider a different group with larger centre. This is the first example of a group where the residual finiteness growth is faster than any polynomial but slower than any exponential.