## The Conjugacy Problem and Higman Embeddings

Citations: | 1 - 0 self |

### BibTeX

@MISC{Ol’shanskii_theconjugacy,

author = {A. Yu. Ol’shanskii and M. V. Sapir},

title = {The Conjugacy Problem and Higman Embeddings},

year = {}

}

### OpenURL

### Abstract

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

### Citations

394 |
Combinatorial Group Theory
- Lyndon, Schupp
- 1977
(Show Context)
Citation Context ...can be embedded into a finitely presented group whose word problem is also in P (a correction to Valiev’s paper was published in Mathematical reviews, review 54 # 413, see also 2 39sLyndon and Schupp =-=[LS]-=- and Manin [Ma]). A simplified version of Valiev’s proof was later published by Aanderaa and Cohen in [AC] (see also Kalorkoti [Kal]). In [BORS], Birget, Rips and the authors of this paper obtained a ... |

112 |
An Introduction to the Theory of Groups
- Rotman
- 1984
(Show Context)
Citation Context ...presentation Z we have got so far by G(M). There are several ways to use the group G(M) to embed G into a finitely presented group. We cannot use the method used by Higman [Hi], Aanderaa [AC], Rotman =-=[Rot]-=-, and us in [BORS] because it leads to problems discovered by Collins in [Col]. We use another method, described in [OlSa2] and used also in [OlSa3]. Consider N − 1 new copies of the trapezium ∆. Numb... |

101 |
Uniqueness theorems for periodic functions
- Fine, Wilf
- 1965
(Show Context)
Citation Context ... desired. Now suppose that there exists a substantial cancellation in the above mentioned sense. Then both words u t and v −t contain a common subword T with |T | ≥ |u| + |v|. This implies (see, e.g. =-=[FW]-=-) that u and v −1 coincide up to a cyclic shift: v −1 ≡ u2u1 and u ≡ u1u2 for some words u1, u2. Furthermore the word wu −1 1 must be freely equal to to a power ul for some l. Therefore utwvt = utulu1... |

50 |
Eqvations in a free group
- Makanin
- 1983
(Show Context)
Citation Context ...e brief history of h and the choice of transition rules in h; the equations correspond to the ages in the brief history and the unknowns are the words wi(z)). It remains to apply a theorem of Makanin =-=[Mak]-=- saying that there exists an algorithm checking if a system of equations over a free group has a solution. The heights of all 70sring computations of this form must be equal by the definition of compr... |

44 |
On group-theoretic decision problems and their classification
- Miller
- 1971
(Show Context)
Citation Context ...can imagine just an ordinary Turing machine which works with words from the free group instead of a free monoid. Here we give just one example of an S-machine which essentially goes back to C. Miller =-=[Mil]-=- (although Miller did not use S-machines). The real S-machine used in this paper is a “descendant” of this S-machine (obtained from Miller’s machine by several “mutations”). Assume that A is a symmetr... |

41 |
Geometry of Defining Relations in Groups
- Ol’shanskii
- 1989
(Show Context)
Citation Context ...lations R = R(S) ∪ R( ¯ S) ∪ {Σ}. 3 The first properties of H 3.1 Diagrams Recall the well-known van Kampen-Lyndon topological interpretation of the consequences of defining relations in groups [LS], =-=[Ol1]-=-. We define a van Kampen diagram over some group presentation L = 〈x1, . . . , xk | P〉 (or briefly, by misuse of language, over the group L), where the defining words from P are cyclically reduced. It... |

40 |
Subgroups of finitely presented groups
- Higman
(Show Context)
Citation Context ...rolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 10 Arrangement of hubs 107 11 The end of the proof 112 References 113 Subject index 116 1 Introduction In 1961, G. Higman =-=[Hi]-=- published the celebrated theorem that a finitely generated group is recursively presented if and only if it is a subgroup of a finitely presented group. Along with the results of Novikov [Nov] and Bo... |

35 | Algorithmic problems in varieties - Kharlampovich, Sapir - 1995 |

33 | Isoperimetric and isodiametric functions of groups
- Sapir, Birget, et al.
- 2002
(Show Context)
Citation Context ...s machine in a group. Of course we would like to have a machine which is easier to interpret. The most suitable machines for our purposes are the so called S-machines invented by the second author in =-=[SBR]-=- and successfully used in [SBR, BORS, OlSa1, OlSa2, OlSa3]. An S-machine works with words which can have complicated structure determined by the problem. Different parts of these words can be elements... |

24 | simple, unsolvable problems of group theory - Boone, Certain - 1957 |

22 | Isoperimetric functions of groups and computational complexity of the word problem
- Ol’shanskii, Sapir
- 2002
(Show Context)
Citation Context ...reviews, review 54 # 413, see also 2 39sLyndon and Schupp [LS] and Manin [Ma]). A simplified version of Valiev’s proof was later published by Aanderaa and Cohen in [AC] (see also Kalorkoti [Kal]). In =-=[BORS]-=-, Birget, Rips and the authors of this paper obtained a group theoretic characterization of NP (non-deterministic polynomial time): a finitely generated group G has word problem in NP if and only if G... |

20 | Embeddings into finitely generated simple groups which preserve the word problem, in Word problems II: The Oxford - Thompson |

12 |
The conjugacy problem and subgroups of finite index
- Collins
- 1977
(Show Context)
Citation Context ...m in NP if and only if G is embedded into a finitely presented group with polynomial Dehn function. The conjugacy problem turned out to be much harder to preserve under embeddings. Collins and Miller =-=[CM]-=- and Gorjaga and Kirkinskiĭ[GK] proved that even subgroups of index 2 of finitely presented groups do not inherit solvability or unsolvability of the conjugacy problem. In 1976 D. Collins [KT] posed t... |

12 |
Length and area functions on groups and quasi-isometric Higman embeddings
- Ol’shanskii, Sapir
(Show Context)
Citation Context ... this machine (that is the machine takes it to the word q) if and only if u = 1 in the group ¯ G, that is if and only u = 1 in G (because G is embedded into ¯ G). 1 1 Notice that by a result of Sapir =-=[OlSa2]-=-, for every Turing machine T there exists an S-machine M corresponding a to a group ¯ G as above which is polynomially equivalent to T. This shows that machines M are as powerful as ordinary Turing ma... |

6 |
An embedding theorem for finitely generated groups
- Clapham
- 1967
(Show Context)
Citation Context ...ne [Bo] this result showed that objects from logic (in that case, recursively enumerable sets) have group theoretic characterizations (see Manin [Ma] for the philosophy of Higman embeddings). Clapham =-=[Cla]-=- (see also corrections in [Va]) was probably the first to investigate properties preserved under Higman embeddings. In particular, he slightly modified the original Higman construction and showed that... |

6 |
The decidability of the conjugacy problem cannot be transferred to finite extensions of groups (Russian). Algebra i Logika 14
- Gorjaga, Kirkinskiĭ
- 1975
(Show Context)
Citation Context ...edded into a finitely presented group with polynomial Dehn function. The conjugacy problem turned out to be much harder to preserve under embeddings. Collins and Miller [CM] and Gorjaga and Kirkinskiĭ=-=[GK]-=- proved that even subgroups of index 2 of finitely presented groups do not inherit solvability or unsolvability of the conjugacy problem. In 1976 D. Collins [KT] posed the following question (problem ... |

6 |
On distortion of subgroups in finitely presented groups
- Ol’shanskii
- 1997
(Show Context)
Citation Context ... word T1 is accepted. The next step is to consider N ≥ 1 copies of our trapezium with labels taken from different alphabets. For technical reasons (similar to the hyperbolic graphs argument in [SBR], =-=[Ol2]-=-, [BORS]) we need N to be even and large enough (say, N ≥ 8). Let us choose disjoint alphabets in different copies of the trapezium. Then we can glue two copies of the trapezium by using a special let... |

6 |
Embeddings of relatively free groups into finitely presented groups, Contemp
- Ol’shanskii, Sapir
(Show Context)
Citation Context ...will change the definition of reduced diagrams by demanding in addition that the top and bottom labels of any θ-band in a reduced diagram are reduced words. 3.4 Forbidden annuli Similarly to [OlSa2], =-=[OlSa1]-=- the absence of annuli of various kinds in reduced diagrams without hubs is important in our present considerations. The following lemma is similar to Lemma 3.1 of [OlSa1]. Lemma 3.11. Let ∆ be a redu... |

5 |
Modular machines and the Higman-Clapham-Valiev embedding theorem, Word problems
- Aanderaa, Cohen
- 1976
(Show Context)
Citation Context ...paper was published in Mathematical reviews, review 54 # 413, see also 2 39sLyndon and Schupp [LS] and Manin [Ma]). A simplified version of Valiev’s proof was later published by Aanderaa and Cohen in =-=[AC]-=- (see also Kalorkoti [Kal]). In [BORS], Birget, Rips and the authors of this paper obtained a group theoretic characterization of NP (non-deterministic polynomial time): a finitely generated group G h... |

5 | Decision problems in group theory
- Kalorkoti
(Show Context)
Citation Context ...thematical reviews, review 54 # 413, see also 2 39sLyndon and Schupp [LS] and Manin [Ma]). A simplified version of Valiev’s proof was later published by Aanderaa and Cohen in [AC] (see also Kalorkoti =-=[Kal]-=-). In [BORS], Birget, Rips and the authors of this paper obtained a group theoretic characterization of NP (non-deterministic polynomial time): a finitely generated group G has word problem in NP if a... |

4 |
Unsolved Problems in Group Theory. 5th edition
- Notebook
- 1976
(Show Context)
Citation Context ... Miller [CM] and Gorjaga and Kirkinskiĭ[GK] proved that even subgroups of index 2 of finitely presented groups do not inherit solvability or unsolvability of the conjugacy problem. In 1976 D. Collins =-=[KT]-=- posed the following question (problem 5.22): Does there exist a version of the Higman embedding theorem in which the degree of unsolvability of the conjugacy problem is preserved? It was quickly real... |

4 |
On the algorithmic insolvability of the word problem in group theory
- Novikov
- 1958
(Show Context)
Citation Context ... Higman [Hi] published the celebrated theorem that a finitely generated group is recursively presented if and only if it is a subgroup of a finitely presented group. Along with the results of Novikov =-=[Nov]-=- and Boone [Bo] this result showed that objects from logic (in that case, recursively enumerable sets) have group theoretic characterizations (see Manin [Ma] for the philosophy of Higman embeddings). ... |

3 |
Conjugacy and the Higman embedding theorem. Word problems
- Collins
- 1976
(Show Context)
Citation Context ...vability of the conjugacy problem is preserved? It was quickly realized that the main problem would be in preserving the smallest Turing degree, that is the solvability of conjugacy problem. In 1980, =-=[Col]-=-, Collins analyzed existing proofs of Higman’s theorem, and discovered that there are essential difficulties. If a finitely generated group C is embedded into a finitely presented group B using any ex... |

2 |
On polynomial reducibility of the word problem under embedding of recursively presented groups in finitely presented groups
- Valiev
- 1975
(Show Context)
Citation Context ... objects from logic (in that case, recursively enumerable sets) have group theoretic characterizations (see Manin [Ma] for the philosophy of Higman embeddings). Clapham [Cla] (see also corrections in =-=[Va]-=-) was probably the first to investigate properties preserved under Higman embeddings. In particular, he slightly modified the original Higman construction and showed that his embedding preserves solva... |

1 |
Algorithms and recursive functions. Translated from the first Russian edition by Leo F. Boron, with the collaboration
- Mal’cev
- 1970
(Show Context)
Citation Context ...teristic function of the set of pairs of words (u, v) which are conjugate in G to the set of elementary recursive functions, and then apply the usual operators used to produce all recursive functions =-=[Mal]-=-. Let us call such functions G-recursive. For example, the word problem in G is G-recursive, by Clapham’s theorem [Cla] the word problem in ¯ G is also G-recursive and so on. The reader can check that... |

1 |
The computable and the non-computable. (Vychislimoe i nevychislimoe
- Manin
(Show Context)
Citation Context ... group. Along with the results of Novikov [Nov] and Boone [Bo] this result showed that objects from logic (in that case, recursively enumerable sets) have group theoretic characterizations (see Manin =-=[Ma]-=- for the philosophy of Higman embeddings). Clapham [Cla] (see also corrections in [Va]) was probably the first to investigate properties preserved under Higman embeddings. In particular, he slightly m... |

1 | Non-amenable finitely presented torsion-by cyclic groups - Olshanskii - 2002 |

1 |
order and conjugacy problems in groups
- Olshanskii, power
(Show Context)
Citation Context ...ions has conjugacy problem of r.e. Turing degree α if and only if G can be Frattini embedded into a finitely presented group H with conjugacy problem of r.e. Turing degree α. In the forthcoming paper =-=[OlSa4]-=-, we will present some corollaries of these theorems (we did not include the proofs of them here in order not to increase the difficulty level unnecessarily). In particular, we will show that one can ... |