On the multiple conjugacy problem in group $F/{N_1\cap N_2}$

O. V. Kulikova

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 163-183.

Résumé

Let $F$ be a free group generated by a finite alphabet $A$. Let $N_1$ ($N_2$) be the normal closure of a finite non-empty symmetrized set $R_1$ (respectively, $R_2$) of elements in $F$. Earlier, one obtained the conditions sufficient for the solvability of the conjugacy problem in the group $F/N_1\cap N_2$. The present paper is a continuation of this research and is devoted to the solvability of the multiple conjugacy problem in $F/{N_1\cap N_2}$. In particular, we get that if $R_1\cup R_2$ satisfies the small cancellation condition $C'(1/6)$, then the multiple conjugacy problem is solvable in $F/{N_1\cap N_2}$.

Bibliographie

[1] Bezverkhnii V. N., Dobrynina I. V., “Reshenie problemy obobschennoi sopryazhennosti slov v gruppakh Kokstera bolshogo tipa”, Diskret. matem., 17:3 (2005), 123–145 | Zbl

[2] Gromov M., Giperbolicheskie gruppy, In-t kompyut. issled, Izhevsk, 2002

[3] Kulikova O. V., “O probleme sopryazhennosti v gruppe $F/{N_1\cap N_2}$”, Matem. zametki, 93:6 (2013), 853–868 | Zbl

[4] Kulikova O. V., Olshanskii A. Yu., “O konechnoi opredelennosti gruppy $F/[M,N]$”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2006, no. 6, 19–21 | MR | Zbl

[5] Lysenok I. G., “O nekotorykh algoritmicheskikh svoistvakh giperbolicheskikh grupp”, Izv. AN SSSR. Ser. matem., 53:4 (1989), 814–832 | Zbl

[6] Olshanskii A. Yu., Geometriya opredelyayuschikh sootnoshenii v gruppakh, Nauka, M., 1989

[7] Sarkisyan R. A., “Problema sopryazhennosti dlya naborov tselochislennykh matrits”, Matem. zametki, 25:6 (1979), 811–824 | MR

[8] Baumslag G., Bridson M. R., Miller C. F., III, Short H., “Fibre products, non-positive curvature and decision problems”, Comm. Math. Helv., 75 (2000), 457–477 | DOI | MR | Zbl

[9] Bogley W. A., Pride S. J., “Aspherical relative presentations”, Proc. Edinburgh Math. Soc., 35:1 (1992), 1–39 | DOI | MR | Zbl

[10] Bridson M. R., Howie J., “Conjugacy of finite subsets in hyperbolic groups”, Int. J. Algebra Comput., 15:4 (2005), 725–756 | DOI | MR | Zbl

[11] Bridson M. R., Howie J., Miller C. F., III, Short H., “On the finite presentation of subdirect products and the nature of residually free groups”, Amer. J. Math., 135 (2013), 891–933 | DOI | MR | Zbl

[12] Bridson M. R., Miller C. F., III, “Structure and finiteness properties of subdirect products of groups”, Proc. London Math. Soc., 98:3 (2009), 631–651 | DOI | MR | Zbl

[13] Collins D. J., “Conjugacy and the Higman embedding theorem”, Word Problems II, Stud. Logic Foundations Math., 95, eds. S. I. Adian, W. W. Boone, G. Higman, North-Holland, Amsterdam, 1980, 81–85 | DOI | MR

[14] Gersten S. M., Short H. B., “Rational subgroups of biautomatic groups”, Ann. Math., 134 (1991), 125–158 | DOI | MR | Zbl

[15] Gersten S. M., Short H. B., “Small cancellation theory and automatic groups”, Invent. Math., 102 (1990), 305–334 | DOI | MR | Zbl

[16] Gutiérrez M. A., Ratcliffe J. G., “On the second homotopy group”, Quart. J. Math. Oxford (2), 32 (1981), 45–55 | DOI | MR | Zbl

[17] Kassabov M., Matucci F., “The simultaneous conjugacy problem in groups of piecewise linear functions”, Groups, Geometry, Dynamics, 6 (2012), 279–315 | DOI | MR | Zbl

[18] Kulikova O. V., “On intersections of normal subgroups in free groups”, Algebra Discrete Math., 2003, no. 1, 36–67 | MR | Zbl

[19] Lindon R. S., Schupp P. E., Combinatorial Group Theory, Springer, Berlin, 1977 | MR

[20] Miller C. F., III, On Group-Theoretic Decision Problems and Their Classification, Ann. Math. Stud., 68, Princeton Univ. Press, 1971 | MR | Zbl

[21] Truiffault B., “Centralisateurs des élé ments dans les groupes de Greendlinger”, C. R. Acad. Sci. Paris Sér. A, 279 (1974), 317–319 | MR

[22] Truiffault B., “Centralisateurs des élé ments d'ordre fini dans les groupes de Greendlinger”, Math. Z., 136 (1974), 7–11 | DOI | MR