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This question concerns right angled Artin groups (RAAGs), also called partially commutative groups or graph groups.

A student of mine, Adi Ben-Zvi, needs for an algorithm in RAAGs, a subalgorithm that we suspect should be known.

In RAAGs, it is known how to compute efficiently the centralizer of a single element [A Baudisch, Subgroups of semifree groups, Acta Math. Acad. Sci. Hungar. 38 (1981) 19–28; H Servatius, Automorphisms of Graph Groups, J. Algebra 126 (1987) 34–60]. In these papers, the cyclically reduced case is treated, but this obviously implies - since we consider a single element - the case of a general element.

Question: Is there an efficient algorithm for computing, in RAAGs, the centralizer of a given finite set of (general) group elements?

We have a positive answer for sets of cyclically reduced elements, but thus far we only have special cases of the general case.

If no such algorithm is known, we would appreciate even results providing information about centralizers of given finite sets.

Update: Following @YCor, by "computing" we mean providing a set of generators (elements of the original RAAG).

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    $\begingroup$ "Computing" a centralizing is not a clearly defined concept: what output should we expect? Actually, we can indeed expect that arbitrary centralizers in RAAGs are finitely generated and hope, as an output, to get generators of the given centralizer. Note that in principle this doesn't say whether two given centralizer are equal! However, RAAGs being linear, we can compute the centralizers in the matrix algebra, and hence determine whether two given finite subsets have equal centralizers. $\endgroup$
    – YCor
    Commented Jun 26, 2018 at 15:59
  • $\begingroup$ @YCor thanks! Doesn't your answer imply there is a (cubic, say) algorithm for the problem? But perhaps pulling back to the RAAG may be difficult? (PS I am ashamed to say that I enjoyed many comments of yours thus far, and still do not know who you are. Would appreciate knowing, if I may.) $\endgroup$
    – Boaz Tsaban
    Commented Jun 26, 2018 at 16:09
  • $\begingroup$ No, I can compute the centralizer in the linear representation (a certain subspace, actually subalgebra, of the matrix algebra), but you still have to determine the intersection of the subgroup with this given subspace, and as I said I don't even know it to be finitely generated and only expect it. $\endgroup$
    – YCor
    Commented Jun 26, 2018 at 16:24

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