2000
DOI: 10.1017/s0305004100004539
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Equations of length 4 and one-relator products

Abstract: This paper continues our study started in [5] of the group G defined by the presentationformula herewhere m [ges ] 2, n [ges ] 2 and ∈i = ±1 for 1 [les ] i [les ] 2. The problem under consideration can be stated as: precisely when does both a have order m and b have order n in the group G? This question arose as a result of our investigation into equations over groups and the connection is explained in Section 2.

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Cited by 13 publications
(9 citation statements)
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“…Freiheitssatz theorems were obtained by Shwartz for Case 1 in [36], for Case 2 in [37], and for Case 3 in [38]; all of these results are contained in [35]. Case 0 was considered in [17], [18] and our arguments below may be applied to this case; however, since these results are more intricate we limit ourselves to applying the results of Cases …”
Section: Freiheitssatz Methods For Prischepov Groupsmentioning
confidence: 99%
See 1 more Smart Citation
“…Freiheitssatz theorems were obtained by Shwartz for Case 1 in [36], for Case 2 in [37], and for Case 3 in [38]; all of these results are contained in [35]. Case 0 was considered in [17], [18] and our arguments below may be applied to this case; however, since these results are more intricate we limit ourselves to applying the results of Cases …”
Section: Freiheitssatz Methods For Prischepov Groupsmentioning
confidence: 99%
“…A one-relator product G = (H * K)/ << R >> (where << R >> denotes the normal closure of R in H * K) is said to satisfy the Freiheitssatz if the natural homomorphisms H → G, K → G are both embeddings. The Freiheitssatz for one-relator products has been considered in many papers -see [17], [18], [26], [38] and the references therein. Setting…”
Section: Amalgamated Free Products and The Freiheitssatzmentioning
confidence: 99%
“…Since H ∼ = Z, if the Freiheitssatz holds in this setting we have that E n (m, k) (and hence G n (m, k)) is infinite. Now R is of the form R = abcd (a, c ∈ H, b, d ∈ K), and the Freiheitssatz for one-relator products where the relator takes this form was studied in [17], [18], [39], [40] and other papers by the same authors. In our proof we will require results from those papers including the following, which we reproduce here as the preprint [39] remains unpublished.…”
Section: The Finite Fibonacci Groups and Sieradski Groupsmentioning
confidence: 99%
“…A series of articles studies the solvability of equations over groups (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein). It is shown there that under certain conditions the equation w(x) = 1 with coefficients in a group G is solvable over G; thus, we can find a group H that includes G as a subgroup and an element h ∈ H satisfying w(h) = 1.…”
Section: Introductionmentioning
confidence: 99%