Residual finiteness properties of some of Halls groups

TL;DR

This paper investigates residual finiteness properties of certain central extensions of wreath products, revealing groups with unique conjugacy and residual finiteness growth behaviors, including examples with intermediate growth rates.

Contribution

It introduces new examples of groups with complex residual finiteness and conjugacy properties, including the first known group with residual finiteness growth between polynomial and exponential.

Findings

Existence of a conjugacy separable group with solvable word problem but unsolvable conjugacy problem.

Construction of groups with conjugacy growth and conjugator length functions of various specified rates.

First example of a group with residual finiteness growth faster than polynomial but slower than exponential.

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 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.