In the article *[Centralizers in R. Thompson group $V\_n$](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-7207&vol=7&iss=4&rank=2)*, the following question is asked:



> 
> **Question:** Is the centraliser of an element of $V\_n$ always finitely presented?
> 
> 
> 


I am wondering: is this question still open? I have a plausible argument in mind, but I would like to be sure that the answer is not already somewhere in the literature.



Centralisers in Thompson's group $V_n$
I finally found the answer to my question. In *[Centralizers in R. Thompson group $V\_n$](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-7207&vol=7&iss=4&rank=2)*, it is proved that the centraliser of an element of $V\_n$ decomposes as 
$$ \left( \prod\limits\_{i=1}^s K\_{m\_i} \rtimes G\_{n,r\_i} \right) \times \left( \prod\limits\_{j=1}^t (( A\_j \rtimes \mathbb{Z} ) \wr P\_{j}) \right)$$
where 


* $A\_1, \ldots, A\_t, P\_{1}, \ldots, P\_t$ are finite groups;
* $K\_{m\_i} = \mathrm{Maps}(\mathfrak{C}\_n, \mathbb{Z}\_{m\_i})^{r\_i}$ where $\mathrm{Maps}(\mathfrak{C}\_n,\mathbb{Z}\_{m\_i})$ is the group of continuous maps from the Cantor set $\mathfrak{C}\_n$ to the cyclic group $\mathbb{Z}\_{m\_i}$ (endowed with the discrete topology) under pointwise multiplication;
* $G\_{n,r\_i}$ is the Higman-Thompson group.


The right factor is virtually $\mathbb{Z}^t$, so no problem here. And it follows from Theorem 4.9 in *[Cohomological finiteness conditions and centralisers in generalisations
of Thompson's group $V$](https://www.degruyter.com/view/j/form.ahead-of-print/forum-2014-0176/forum-2014-0176.xml)* that the left factor is of type $F\_\infty$ (i.e., it admits a classifying space containing finitely many cells in each dimension). 


It follows that centralisers in $V\_n$ are not only finitely presented, they are of type $F\_\infty$. 


