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Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$?

Note that a subgroup $H$ of $G$ is malnormal if $H^g\cap H\neq 1$ implies that $g\in H$. That is, $H$ intersects each of its proper conjugates trivially.

This seems far to strong (far too good!) to be true. However, I cannot find anything in the literature that even hints at the existence or non-existence of such an embedding.

(More generally, I am wondering if there is some kinds of embedding which preserves the centraliser of a given element of $Q$. That is, if we fix $x\in Q$ then can $Q$ be embedded in a finitely presented group $G$ such that $C_Q(x)=C_G(x)$ (or even $C_Q(x)\cong C_G(x)$)? Malnormality guarantees this, and is a more interesting question, but I thought I should mention this too...)

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3 Answers 3

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This is solved in my paper here using a generalized version of the techniques of Sapir. We also show some other properties of the embedding can be achieved while adding malnormality, e.g that the embedding is a q-i and preserves the decidability of the Word problem.

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    $\begingroup$ Nice results, well done! $\endgroup$
    – Ashot Minasyan
    Commented Apr 30 at 9:34
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This appears to be an open question: see Remark 5.23 of Sapir's paper, which attributes this as a question to Denis Osin.

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As I wrote in Remark 5.23 cited by Ian Agol, the embedding from that paper should answer the first question. It certainly preserves all centralizers, that is basically proved in the paper. That is if $G$ is a finitely generated recursively presented group, $Q>G$ is the group constructed in that paper, then for every $x\ne 1$ in $G$, $C_Q(x)=C_G(x)$. In fact the version of Higman embedding in Olʹshanskii, A. Yu.; Sapir, M. V. The conjugacy problem and Higman embeddings. Mem. Amer. Math. Soc. 170 (2004), no. 804 satisfies this property too. If I had more time, I would list lemmas from that paper which imply that. I am not sure if that embedding answers your first question. It might.

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  • $\begingroup$ I am struggling to find where you prove that the centralisers are preserved in your "aspherical groups..." paper. Could you perhaps point me in the right direction? Also, having come back to this after a while I have realised that the two constructions you give only work for finitely-generated groups. Do you know if it is "not impossible" for arbitrary recursively presented groups? (I suppose I am asking: where does the limitation of finitely generated come from? Is it just the techniques, or is it more fundamental than that?) $\endgroup$
    – ADL
    Commented May 31, 2013 at 12:50
  • $\begingroup$ I am writing a paper about it, should be ready in a couple of weeks. $\endgroup$
    – user6976
    Commented May 31, 2013 at 14:45
  • $\begingroup$ Oh, that is excellent. I will look forward to it! $\endgroup$
    – ADL
    Commented May 31, 2013 at 17:07

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