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The conjugacy problem for groups, and Higman embeddings
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## The conjugacy problem for groups, and Higman embeddings

### A. Yu. Ol'shanskii and M. V. Sapir

**Abstract.**
For every finitely generated recursively
presented group ${\mathcal G}$
we construct a finitely presented group ${\mathcal H}$
containing ${\mathcal G}$ such that ${\mathcal G}$ is (Frattini) embedded into
${\mathcal H}$ and the group ${\mathcal H}$
has solvable conjugacy problem if and
only if ${\mathcal G}$ has solvable conjugacy problem. Moreover, ${\mathcal G}$ and
${\mathcal H}$ have the same r.e.~Turing degrees of the conjugacy problem.
This solves a problem by D. Collins.

*Copyright 2003 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**09** (2003), pp. 40-50
- Publisher Identifier: S 1079-6762(03)00110-0
- 2000
*Mathematics Subject Classification*. Primary 20F10; Secondary 03D40, 20M05
- Received by editors March 2, 2003
- Posted on June 24, 2003
- Communicated by Efim Zelmanov
- Comments (When Available)

**A. Yu. Ol'shanskii**

Mathematics Department, Vanderbilt University, Nashville, Tennessee 37240, and Mechanics-Mathematics Department, Chair of Higher Algebra, Moscow State University, Moscow, Russia

*E-mail address:* `alexander.olshanskiy@vanderbilt.edu, olshan@shabol.math.msu.su `

**M. V. Sapir**

Mathematics Department, Vanderbilt University, Nashville, Tennessee 37240

*E-mail address:* `msapir@math.vanderbilt.edu`

Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 02-01-00170 and by the INTAS grant 99-1224; the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298.

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