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Generalized Fibonacci groups \(H(r,n,s)\) that are connected labelled oriented graph groups. (English) Zbl 1439.20050

A labelled oriented graph (LOG) consists of a finite graph (possibly with loops and multiple edges) with vertex set \(V\) and edge set \(E\) together with three maps \(\iota, \tau, \lambda: \; E \rightarrow V\). A LOG determines a group presentation \(\langle V \mid \lambda(e)^{-1}\iota(e)\lambda(e)=\tau(e), \; e \in E \rangle\) and a group with a LOG presentation is called a LOG group. The class of connected LOG groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in \(S^{4}\), and so contains all knot groups.
This article is devoted to the study of generalized Fibonacci groups: \( H(r,n,s)= \langle x_{0},x_{1}, \ldots x_{n-1} \mid \prod_{j=0}^{r-1}x_{i+j}=\prod_{j=0}^{s-1}x_{i+j+r}, \;\; 0 \leq j <n \rangle \) where \(r,s \geq 1\), \(n \geq 2\) and subscripts are taken \(\bmod n\). The main result is Theorem A: If \(H(r,n,s)\) is a connected LOG group, then one of the following holds: (a) \(r=s\) and \(\mathrm{GCD}(r,n)=1\) in which case \(H(r,n,s) \simeq \langle a,b \mid a^{n}=b^{r} \rangle\) is the fundamental group of the complement of the \((r,n)\)-torus knot in \(S^{3}\); (b) \(\mathrm{GCD}(r,n,s)=2\) and either \(r \equiv 0 \mod n\) or \(s \equiv 0 \mod n\), in which cases \(H(r,n,s) \simeq \mathbb{Z}\); (c) \(\vert r-s \vert=2\), \(\{r,s\} \not = \{2,4\}\), \(r \not \equiv 0 \mod n\), \(s \not \equiv 0 \mod n\), and the group \(H(r/2,n/2,s/2)\) is perfect. Furthermore, it is conjectured that case (c) cannot be given.
Let \(\beta\) be the Betti number (or torsion-free rank) of the abelianization of \(H(r,n,s)\). Theorem C: Let \(r,s \geq 1\), \(n \geq 2\). Then (a) if \(r \not =s\), then \(\beta=\mathrm{GCD}(r,n,s)-1\); (b) if \(r=s\), then \(b=\mathrm{GCD}(r,n)\).

MSC:

20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
57M07 Topological methods in group theory
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C20 Directed graphs (digraphs), tournaments

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References:

[1] S. A. Bleiler, Two-generator cable knots are tunnel one, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1285-1287. · Zbl 0841.57008
[2] S. A. Bleiler and A. C. Jones, On two generator satellite knots, Geom. Dedicata 104 (2004), 1-14. · Zbl 1053.57001
[3] W. A. Bogley and G. Williams, Efficient finite groups arising in the study of relative asphericity, Math. Z. 284 (2016), no. 1-2, 507-535. · Zbl 1397.20036
[4] W. A. Bogley and G. Williams, Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups, J. Algebra 480 (2017), 266-297. · Zbl 1371.20036
[5] A. M. Brunner, On groups of Fibonacci type, Proc. Edinb. Math. Soc. (2) 20 (1976/77), no. 3, 211-213. · Zbl 0386.20016
[6] C. M. Campbell and E. F. Robertson, On a class of finitely presented groups of Fibonacci type, J. Lond. Math. Soc. (2) 11 (1975), no. 2, 249-255. · Zbl 0309.20011
[7] C. M. Campbell and R. M. Thomas, On infinite groups of Fibonacci type, Proc. Edinb. Math. Soc. (2) 29 (1986), no. 2, 225-232. · Zbl 0595.20028
[8] A. Cavicchioli, D. Repovš and F. Spaggiari, Topological properties of cyclically presented groups, J. Knot Theory Ramifications 12 (2003), no. 2, 243-268. · Zbl 1031.57002
[9] J. H. Conway, Advanced problem 5327, Amer. Math. Monthly 72 (1965), 915-915.
[10] J. H. Conway, J. A. Wenzel, R. C. Lyndon and H. Flanders, Problems and solutions: Solutions of advanced problems: 5327, Amer. Math. Monthly 74 (1967), no. 1, 91-93.
[11] J. E. Cremona, Unimodular integer circulants, Math. Comp. 77 (2008), no. 263, 1639-1652. · Zbl 1217.11028
[12] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.6, 2016.
[13] N. D. Gilbert, Labelled oriented graph groups and crossed modules, Arch. Math. (Basel) 108 (2017), no. 4, 365-371. · Zbl 1388.20069
[14] N. D. Gilbert and J. Howie, LOG groups and cyclically presented groups, J. Algebra 174 (1995), no. 1, 118-131. · Zbl 0851.20025
[15] N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford Sci. Publ., The Clarendon Press, New York, 1994. · Zbl 0828.57001
[16] F. González-Acuña, C. McA. Gordon and J. Simon, Unsolvable problems about higher-dimensional knots and related groups, Enseign. Math. (2) 56 (2010), no. 1-2, 143-171. · Zbl 1213.57004
[17] J. Harlander and S. Rosebrock, Aspherical word labeled oriented graphs and cyclically presented groups, J. Knot Theory Ramifications 24 (2015), no. 5, Article ID 1550025. · Zbl 1373.57007
[18] J. Howie, On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), no. 1, 281-302. · Zbl 0572.57001
[19] J. Howie and G. Williams, Tadpole labelled oriented graph groups and cyclically presented groups, J. Algebra 371 (2012), 521-535. · Zbl 1275.20027
[20] A. W. Ingleton, The rank of circulant matrices, J. Lond. Math. Soc. 31 (1956), 445-460. · Zbl 0072.00802
[21] D. L. Johnson, Topics in the Theory of Group Presentations, London Math. Soc. Lecture Note Ser. 42, Cambridge University Press, Cambridge, 1980. · Zbl 0437.20026
[22] D. L. Johnson, J. W. Wamsley and D. Wright, The Fibonacci groups, Proc. Lond. Math. Soc. (3) 29 (1974), 577-592. · Zbl 0327.20014
[23] L. P. Neuwirth, Knot Groups, Ann. of Math. Stud. 56, Princeton University Press, Princeton, 1965. · Zbl 0184.48903
[24] M. Newman, Circulants and difference sets, Proc. Amer. Math. Soc. 88 (1983), no. 1, 184-188. · Zbl 0516.05013
[25] M. Ozawa, Knots and surfaces, Sūgaku 67 (2015), no. 4, 403-423.
[26] M. I. Prishchepov, Asphericity, atoricity, and symmetrically presented groups, Comm. Algebra 23 (1995), no. 13, 5095-5117. · Zbl 0861.20036
[27] M. Scharlemann, Tunnel number one knots satisfy the Poenaru conjecture, Topology Appl. 18 (1984), no. 2-3, 235-258. · Zbl 0592.57004
[28] J. Simon, Wirtinger approximations and the knot groups of F^{n} in S^{n+2}, Pacific J. Math. 90 (1980), no. 1, 177-190. · Zbl 0461.57008
[29] J. R. Stallings, On the recursiveness of sets of presentations of 3-manifold groups, Fund. Math. 51 (1962), 191-194. · Zbl 0121.40006
[30] A. Szczepański and A. Vesnin, On generalized Fibonacci groups with an odd number of generators, Comm. Algebra 28 (2000), no. 2, 959-965. · Zbl 0951.20023
[31] A. Szczepański and A. Vesnin, HNN extension of cyclically presented groups, J. Knot Theory Ramifications 10 (2001), no. 8, 1269-1279. · Zbl 1011.20026
[32] G. Williams, Largeness and SQ-universality of cyclically presented groups, Internat. J. Algebra Comput. 22 (2012), no. 4, Article ID 1250035. · Zbl 1269.20026
[33] G. Williams, Fibonacci type semigroups, Algebra Colloq. 21 (2014), no. 4, 647-652. · Zbl 1311.20036
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