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Hyperbolic alternating virtual link groups. (English) Zbl 1244.57008
The article under review studies the geometry of certain link complements. The author identifies two types of forbidden tangles and proves that if a prime, alternating link projection does not contain either of those two tangles then the fundamental group \(G\) of the complement is the fundamental group of a finite, piecewise Euclidean 2-complex of nonpositive curvature. If one assumes that the link projection is dense, then \(G\) is shown to be hyperbolic.

57M05 Fundamental group, presentations, free differential calculus
57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
Full Text: DOI
[1] J. Barnard and N. Brady, Distortion of surface groups in CAT.0/ free-by-cyclic groups. Geom. Dedicata 120 (2006), 119-139. · Zbl 1167.20024 · doi:10.1007/s10711-006-9072-1
[2] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), 445-470. · Zbl 0888.20021 · doi:10.1007/s002220050168
[3] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature . Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999. · Zbl 0988.53001
[4] W. A. Bogley, J.H.C. Whitehead’s asphericity question. In Two-dimensional homotopy and combinatorial group theory , London Math. Soc. Lecture Note Ser. 197, Cambridge University Press, Cambridge 1993, 309-334. · Zbl 0811.57008
[5] J. Harlander and S. Rosebrock, Generalized knot complements and some aspher- ical ribbon disc complements. J. Knot Theory Ramifications 12 (2003), 947-962. · Zbl 1053.57005 · doi:10.1142/S0218216503002871
[6] J. Howie, Some remarks on a problem of J. H. C. Whitehead. Topology 22 (1983), 475-485. · Zbl 0524.57002 · doi:10.1016/0040-9383(83)90038-1
[7] J. Howie, On the asphericity of ribbon disc complements. Trans. Amer. Math. Soc. 289 (1985), 281-302. · Zbl 0572.57001 · doi:10.2307/1999700
[8] G. Huck and S. Rosebrock, Aspherical labelled oriented trees and knots. Proc. Edinb. Math. Soc. (2) 44 (2001), 285-294. · Zbl 0983.57003 · doi:10.1017/S0013091599000474
[9] G. Huck and S. Rosebrock, Spherical diagrams and labelled oriented trees. Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 519-530. · Zbl 1134.57002 · doi:10.1017/S0308210505000053
[10] M. Kapovich, Hyperbolic manifolds and discrete groups . Progr. Math. 183, Birkhäuser, Boston 2001. · Zbl 0958.57001
[11] L. H. Kauffman, Virtual knot theory. European J. Combin. 20 (1999), 663-690. · Zbl 0938.57006 · doi:10.1006/eujc.1999.0314 · arxiv:math/9811028
[12] J. McCammond, Constructing non-positively curved spaces. In Geometric and cohomo- logical methods in group theory , London Math. Soc. Lecture Note Ser. 358, Cambridge University Press, Cambridge 2009, 162-224. · Zbl 1237.20035 · arxiv:1411.7306
[13] W. Menasco, Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), 37-44. · Zbl 0525.57003 · doi:10.1016/0040-9383(84)90023-5
[14] J. W. Morgan, On Thurston’s uniformization theorem for three-dimensional manifolds. In The Smith conjecture , Pure Appl. Math. 112, Academic Press, Orlando 1984, 37-125. · Zbl 0599.57002
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