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This appears to be an open question: see Remark 5.23 of Sapir's paperRemark 5.23 of Sapir's paper, which attributes this as a question to Denis Osin.

This appears to be an open question: see Remark 5.23 of Sapir's paper, which attributes this as a question to Denis Osin.

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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Ian Agol
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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.