Geodesic
A Generalization of Fibonacci Groups
Algebra i logika, Tome 42 (2003) no. 2, pp. 131-160
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We study the class of cyclically presented groups which contain Fibonacci groups and Sieradski groups. Conditions are specified for these groups to be finite, pairwise isomorphic, or aspherical. As a partial answer to the question of Cavicchioli, Hegenbarth, and Repov, it is stated that there exists a wide subclass of groups with an odd number of generators cannot appear as fundamental groups of hyperbolic three-dimensional manifolds of finite volume. 
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V. G. Bardakov; A. Yu. Vesnin. A Generalization of Fibonacci Groups. Algebra i logika, Tome 42 (2003) no. 2, pp. 131-160. http://geodesic.mathdoc.fr/item/AL_2003_42_2_a0/

[1] A. Cavicchioli, F. Hegenbarth, D. Repovš, “On manifold spines and cyclic presentations of groups”, Knot Theory, Banach Cent. Publ., 42, PWN, Warsaw, 1998, 49–56 | MR

[2] U. Massi, Dzh. Stollings, Algebraicheskaya topologiya, Mir, M., 1977 | MR | Zbl

[3] D. Kollinz, Kh. Tsishang, “Kombinatornaya teoriya grupp i fundamentalnye gruppy”, Algebra-7, Itogi nauki i tekhniki. Sovrem. probl. matem. Fundam. napravleniya, 58, VINITI, M., 1990, 5–190 | MR

[4] J. Stallings, “On the recursiveness of sets of presentations of 3-manifold groups”, Fundam. Math., 51 (1962/63), 191–194 | MR | Zbl

[5] J. Conway, “Advanced problem 5327”, Am. Math. Mon., 72 (1965), 915 | DOI | MR

[6] D. L. Johnson, Topics in the theory of group presentations, Lond. Math. Soc. Lect. Note Ser., 42, Cambridge Univ. Press, Cambridge, 1980 | MR | Zbl

[7] D. L. Johnson, J. W. Wamsley, D. Wright, “The Fibonacci Groups”, Proc. London Math. Soc., III. Ser., 29:4 (1974), 577–592 | DOI | MR | Zbl

[8] R. Thomas, “The Fibonacci groups $F(2,2m)$”, Bull. London Math. Soc., 21:5(92) (1989), 463–465 | DOI | MR | Zbl

[9] R. Thomas, “The Fibonacci groups revised”, Geometry of Banach spaces, Proc. conf. held in Strobol, Austria, Lond. Math. Soc. Lect. Note Ser., 160, Cambridge Univ. Press, Cambridge, 1989, 445–456 | MR

[10] A. Cavicchioli, F. Spaggiari, “The classification of 3-manifolds with spines related to Fibonacci groups”, Algebraic topology. Homotopy and group homology, Lect. Notes Math., 1509, Springer-Verlag, Berlin a. o., 1992, 50–78 | MR

[11] M. J. Dunwoody, “Cyclic presentations and 3-manifolds”, Proc. inter. conf. “Groups-Korea'94”, Walter de Gruyter, Berlin–New York, 1995, 47–55 | MR | Zbl

[12] A. C. Kim, “On the Fibonacci group and related topics”, Proc. conf. algebra (1991, Barnaul, Russia), Contemp. Math., 184, Am. Math. Soc, Providence, RI, 1995, 231–235 | MR | Zbl

[13] A. C. Kim, Y. Kim, A. Vesnin, Proc. inter. conf. “Groups-Korea 98”, Walter de Gruyter, Berlin–New York, 2000 | Zbl

[14] G. Kim, Y. Kim, A. Vesnin, “The knot $5_2$ and cyclically presented groups”, J. Korean Math. Soc., 35 (1998), 961–980 | MR | Zbl

[15] A. Yu. Vesnin, A. Ch. Kim, “Drobnye gruppy Fibonachchi i mnogoobraziya”, Sib. matem. zhurnal, 39:4 (1998), 765–775 | MR | Zbl

[16] C. Maclachlan, A. W. Reid, “Generalised Fibonacci manifolds”, Transform. Groups, 2:2 (1997), 165–182 | DOI | MR | Zbl

[17] M. Takahashi, “On the presentations of the fundamental groups of 3-manifolds”, Tsukuba J. Math., 13 (1989), 175–189 | MR | Zbl

[18] P. Bandieri, A. C. Kim, M. Mulazzani, “On the cyclic coverings of the knot $5_2$”, Proc. Edinb. Math. Soc., II. Ser., 42:3 (1999), 575–587 | DOI | MR | Zbl

[19] M. R. Casali, “Fundamental groups of branched covering spaces of $S^3$”, Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 33 (1987), 247–258 | MR | Zbl

[20] A. Cavicchioli, R. Ruini, F. Spaggiari, “Cyclic branched coverings of 2-bridge knots”, Revista Mat. Complut., 12 (1999), 383–416 | MR | Zbl

[21] H. Helling, A. C. Kim, J. L. Mennicke, “A geometric study of Fibonacci groups”, J. Lie Theory, 8 (1998), 1–23 | MR | Zbl

[22] H. Helling, A. C. Kim, J. L. Mennicke, “Some honey-combs in hyperbolic 3-space”, Commun. Algebra, 23:14 (1995), 5169–5206 | DOI | MR | Zbl

[23] J. Minkus, “The branched cyclic coverings of 2 bridge knots and links”, Mem. Am. Math. Soc., 35:255 (1982), 1–68 | MR

[24] C. Maclachlan, “Generalizations of Fibonacci numbers, groups and manifolds”, Combinatorial and geometric group theory, Edinburg, 1993, Lond. Math. Soc. Lect. Note Ser., 204, 1995, 233–238 | MR | Zbl

[25] M. I. Prischepov, “Asphericity, atoricity, and symmetrically presented groups”, Commun. Algebra, 23:13 (1995), 5095–5117 | DOI | MR

[26] A. Sieradski, “Combinatorial squashings, 3-manifolds, and the third homology of groups”, Invent. Math., 84:1 (1986), 121–139 | DOI | MR | Zbl

[27] A. Cavicchioli, F. Hegenbarth, A. C. Kim, “A geometric study of Sieradski groups”, Algebra Colloq., 5:2 (1998), 203–217 | MR | Zbl

[28] R. Thomas, “On a question of Kim concerning certain group presentations”, Bull. Korean Math. Soc., 28 (1991), 219–244 | MR

[29] N. Gilbert, J. Howie, “LOG groups and cyclically presented groups”, J. Algebra, 174:1 (1995), 118–131 | DOI | MR | Zbl

[30] R. W. K. Odoni, “Some Diophantine problems arising in the theory of cyclically presented groups”, Glasg. Math. J., 41:2 (1999), 157–166 | DOI | MR

[31] M. Edjvet, “On the asphericity of one-relator relative presentations”, Proc. R. Soc. Edinb., Sect. A, Math., 124 (1994), 713–728 | MR | Zbl

[32] GAP: Groups, algorithms and programming, http://www.dcs.st-and.ac.uk/gap

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

[34] A. Yu. Olshanskii, Geometriya opredelyayuschikh sootnoshenii v gruppakh, Nauka, M., 1989 | MR

[35] C. P. Chalk, “Fibonacci groups with aspherical presentations”, Commun. Algebra, 26:5 (1998), 1511–1546 | DOI | MR | Zbl

[36] A. Szczepański, A. Vesnin, “On generalized Fibonacci groups with odd number of generators”, Commun. Algebra, 28:2 (2000), 959–965 | DOI | MR | Zbl

[37] P. Skott, Geometrii na trekhmernykh mnogoobraziyakh, Mir, M., 1986

[38] A. Berdon, Geometriya diskretnykh grupp, Nauka, M., 1986 | MR

[39] E. B. Vinberg, O. V. Shvartsman, “Diskretnye gruppy dvizhenii prostranstv postoyannoi krivizny”, Geometriya-2, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napravleniya, 29, VINITI, M., 1988, 147–259

[40] J. L. Mennicke, Groups-Korea 1988, Lect. Note Math., 1398, 1988 | MR