Doodle groups. (English) Zbl 0869.57001
Summary: A doodle is a finite number of closed curves without triple intersections on an oriented surface. There is a “fundamental” group, naturally associated with a doodle. In this paper we study these groups, in particular, we show that fundamental groups of some doodles are automatic and give examples of doodles whose fundamental groups have non-trivial center.
MSC:
| 57M05 | Fundamental group, presentations, free differential calculus |
References:
| [1] | Anders Björner and Volkmar Welker, The homology of ”\?-equal” manifolds and related partition lattices, Adv. Math. 110 (1995), no. 2, 277 – 313. · Zbl 0845.57020 · doi:10.1006/aima.1995.1012 |
| [2] | David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. · Zbl 0764.20017 |
| [3] | Roger Fenn and Paul Taylor, Introducing doodles, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 37 – 43. · Zbl 0409.57003 |
| [4] | Roger A. Fenn, Techniques of geometric topology, London Mathematical Society Lecture Note Series, vol. 57, Cambridge University Press, Cambridge, 1983. · Zbl 0517.57001 |
| [5] | S. M. Gersten and H. B. Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990), no. 2, 305 – 334. · Zbl 0714.20016 · doi:10.1007/BF01233430 |
| [6] | S. M. Gersten and H. Short, Small cancellation theory and automatic groups. II, Invent. Math. 105 (1991), no. 3, 641 – 662. · Zbl 0734.20014 · doi:10.1007/BF01232283 |
| [7] | M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3 |
| [8] | M. Khovanov, Real \(K(\pi ,1)\) arrangements from finite root systems, Math. Res. Let., 3 (1996), 261-274. CMP96:11 · Zbl 0861.55010 |
| [9] | G. B. Shabat and V. A. Voevodsky, Drawing curves over number fields, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 199 – 227. · Zbl 0790.14026 · doi:10.1007/978-0-8176-4576-2_8 |
| [10] | V. Voevodsky, Flags and Grothendieck cartographical group in higher dimensions, CSTARCI Math. Preprint 05-90, Moscow, 1990. |
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