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Let $G$ be a virtually compact special group, in the terminology of Haglund and Wise (i.e., $G$ has a finite index subgroup $H$ which is isomorphic to the fundamental group of some compact special non-positively curved cube complex $X$).

Question: for any element $g \in G$, is the centralizer $C_G(g)$ also a virtually compact special group?

From the work of Haglund and Wise we know that the answer is positive if $G$ is hyperbolic, so I am actually asking about the non-hyperbolic case.

In general, some properties of $C_G(g)$ can be deduced from the fact that $G$ is semihyperbolic (in particular, the CAT$(0)$-quasiconvexity), but I am not sure whether these are helpful for the question.

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  • $\begingroup$ You can probably use Servatius' Centralizer Theorem to check that centralizers in RAAGs are quasiconvex. It then follows for compact special groups just because an intersection of quasiconvex subgroups is quasiconvex. I haven't thought about how to extend this to virtually compact special groups. $\endgroup$
    – HJRW
    Dec 26, 2014 at 16:51

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