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In the article Centralizers in R. Thompson group $V_n$, 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.

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  • $\begingroup$ You can ask Collin Bleak, he certainly knows the answer to your question. Alternatively use google scholar to find all texts on the internet referring to this paper. $\endgroup$
    – user6976
    Commented May 1, 2019 at 14:30

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I finally found the answer to my question. In Centralizers in R. Thompson group $V_n$, 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 $V$ 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 $F_\infty$.

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  • $\begingroup$ It seems that you're using two different notations in the same post for $V_n$ (which is among the $G_{n,m}$). I guess that all you say applies for centralizers in arbitrary Higman-Thompson groups (the restriction to $V_n$ is not natural). $\endgroup$
    – YCor
    Commented May 2, 2019 at 17:06
  • $\begingroup$ The decomposition of centralisers is stated and proved only for $V_n$. It seems reasonable to think that the argument also applies to Higman-Thompson groups, but I prefer to refer to a statement which is actually proved. $\endgroup$
    – AGenevois
    Commented May 2, 2019 at 18:41
  • $\begingroup$ OK I see, it would require checking whether any specific thing is used ($V_n$ is just one particular choice among the $n-1$ Higman-Thompson groups $V_{n,m}$ for $m\in\mathbf{Z}/(n-1)\mathbf{Z}$). $\endgroup$
    – YCor
    Commented May 2, 2019 at 18:53

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