On the conjugacy problem in the group \(F/N_1\cap N_2\). (English. Russian original) Zbl 1275.20029
Math. Notes 93, No. 6, 837-849 (2013); translation from Mat. Zametki 93, No. 6, 853-868 (2013).
Summary: Let \(N_1\) (\(N_2\)) be the normal closure of a finite symmetrized set \(R_1\) (\(R_2\), respectively) in a finitely generated free group \(F=F(A)\). As is known, if \(R_i\) satisfies condition \(C(6)\), then the conjugacy problem is decidable in \(F/N_i\). In the paper, it is proved that, if one adds to condition \(C(6)\) on the set \(R_1\cup R_2\) the atoricity condition for the presentation \(\langle A\mid R_1,R_2\rangle\), then the conjugacy problem is decidable in the group \(F/N_1\cap N_2\) as well. In particular, for the decidability of the conjugacy problem in \(F/N_1\cap N_2\), it is sufficient to assume that the set \(R_1\cup R_2\) satisfies condition \(C(7)\).
MSC:
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20F06 | Cancellation theory of groups; application of van Kampen diagrams |
| 20E05 | Free nonabelian groups |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
finitely generated free groups; conjugacy problem; finite symmetrized sets in free groups; presentations; atoricity condition; condition \(C(6)\); condition \(C(7)\); subdirect products; small cancellation conditionsReferences:
| [1] | M. R. Bridson, C. F. Miller III, “Structure and finiteness properties of subdirect products of groups”, Proc. London Math. Soc. (3), 98:3 (2009), 631 – 651 · Zbl 1167.20016 · doi:10.1112/plms/pdn039 |
| [2] | C. F. Miller III, On Group-Theoretic Decision Problems and Their Classification, Ann. Math. Stud., 68, Princeton Univ. Press, Princeton, NJ, 1971 · Zbl 0277.20054 |
| [3] | G. Baumslag, M. R. Bridson, C. F. Miller III, H. Short, “Fibre products, non-positive curvature, and decision problems”, Comment. Math. Helv., 75:3 (2000), 457 – 477 · Zbl 0973.20034 · doi:10.1007/s000140050136 |
| [4] | K. Igusa, The Generalized Grassmann Invariant, Preprint, Brandeis Univ., Waltham, MA, 1979 |
| [5] | C. P. Rourke, “Presentations and the trivial group”, Topology of Low-dimensional Manifolds, Lecture Notes in Math., 722, Springer-Verlag, Berlin, 1979, 134 – 143 · Zbl 0408.57001 · doi:10.1007/BFb0063195 |
| [6] | S. J. Pride, “Identities among relations of group presentations”, Group Theory from a Geometrical Viewpoint, World Sci. Publ., River Edge, NJ, 1991, 687 – 717 · Zbl 0843.20026 |
| [7] | W. A. Bogley, S. J. Pride, “Calculating Generators of \(\pi_2\)”, Two-dimensional Homotopy Theory and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser., 197, Cambridge Univ. Press, Cambridge, 1993, 157 – 188 · Zbl 0811.57005 |
| [8] | R. S. Lyndon, P. E. Schupp, Combinatorial Group Theory, Ergeb. Math. Grenzgeb., 89, Springer-Verlag, Berlin, 1977 · Zbl 0368.20023 |
| [9] | А. Ю. Ольшанский, Геометрия определяющих соотношений в группах, Современная алгебра, Наука, М., 1989 · Zbl 0676.20014 |
| [10] | M. A. Gutieŕrez, J. G. Ratcliffe, “On the second homotopy group”, Quart. J. Math. Oxford Ser. (2), 32:1 (1981), 45 – 55 · Zbl 0457.57002 · doi:10.1093/qmath/32.1.45 |
| [11] | O. V. Kulikova, “On intersections of normal subgroups in free groups”, Algebra Discrete Math., 2003, no. 1, 36 – 67 · Zbl 1164.20342 |
| [12] | Н. В. Безверхний, “Разрешимость проблемы вхождения в циклическую подгруппу в группе с условием \(C(6)\)”, Фундамент. и прикл. матем., 5:1 (1999), 39 – 46 · Zbl 0970.20021 |
| [13] | W. A. Bogley, S. J. Pride, “Aspherical relative presentations”, Proc. Edinburgh Math. Soc. (2), 35:1 (1992), 1 – 39 · Zbl 0802.20029 · doi:10.1017/S0013091500005290 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
