Residual finiteness properties of some of Halls groups
Lukas Vandeputte
TL;DR
The paper analyzes central quotients of Hall's group $G_0$ with large centres to quantify residual finiteness and conjugacy phenomena via growth invariants $Rf_{G,S}$, $Conj_{G,S}$ and $Cl_{G,S}$. It develops a framework linking these invariants to the periodicity of central-defining data $d(i)$ and constructs explicit examples in the class $\mathscr{H}$, including a finitely generated recursively presented, conjugacy separable group with solvable word problem but unsolvable conjugacy problem. It also demonstrates that conjugacy-separability growth can be arbitrarily large while the conjugator length remains polynomial, and that residual finiteness growth can take intermediate or even super-polynomial forms in groups with enlarged centres. Finally, it shows the existence of a group with intermediate residual finiteness growth by building a group with a centre of large rank and a carefully engineered filtration, thereby establishing a rich landscape of independent behaviors among $Rf$, $Conj$, and $Cl$ within Hall-type solvable groups.
Abstract
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 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 $f$. 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.