Residual finiteness properties of some of Halls groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918299221098496 |
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| author | Vandeputte, Lukas |
| author_facet | Vandeputte, Lukas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residual finiteness properties of some of Halls groups Vandeputte, Lukas Group Theory 20F10, 20F65, 20F69 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. |
| title | Residual finiteness properties of some of Halls groups |
| topic | Group Theory 20F10, 20F65, 20F69 |
| url | https://arxiv.org/abs/2508.16452 |