Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A group of paths in $% \mathbb {R}^2$

Author: Richard Kenyon
Journal: Trans. Amer. Math. Soc. 348 (1996), 3155-3172
MSC (1991): Primary 20F34, 20E08, 58F03
MathSciNet review: 1340179
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define a group structure on the set of compact ``minimal'' paths in $\mathbb {R} ^2$. We classify all finitely generated subgroups of this group $G$: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in $G$. The subgroups of $G$ isomorphic to surface groups arise from certain topological $1$-forms on the corresponding surfaces. We construct examples of such $1$-forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using $G$ we construct a non-polygonal tiling problem in $\mathbb {R} ^2$, that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group $G$ has applications to combinatorial tiling problems of the type: given a set of tiles $P$ and a region $R$, can $R$ be tiled by translated copies of tiles in $P$?

References [Enhancements On Off] (What's this?)

  • 1. P. Arnoux, A. Fathi; Un exemple de difféomorphisme pseudo-Anosov. C. R. Acad. Sci. Paris 312 I, (1991):214-244. MR 91j:58122
  • 2. P. Arnoux, G. Levitt; Sur l'unique ergodicité des $1$-formes fermées singulières. Invent. Math. 84 (1986) 141-156. MR 87g:58004
  • 3. M. Bestvina, M. Feighn; Stable actions of groups on real trees, Invent. Math. 121 (1995), 287--321.
  • 4. F. Bonahon; Geodesic laminations with transverse Hölder distributions. preprint.
  • 5. J. H. Conway, J. Lagarias; Tilings with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A. 53 (1990):183-206. MR 91a:05030
  • 6. F. M. Dekking; Recurrent Sets. Adv. in Math. 44 (1982) 78-104. MR 84e:52023
  • 7. A. Fathi; Some compact invariant sets for hyperbolic linear automorphisms of tori. Erg. Thy. Dyn. Syst. 8 (1988):191-204. MR 90j:58116
  • 8. A. Fathi, F. Laudenbach, V. Poenaru; Travaux de Thurston sur les surfaces, Asterisque 66-67 (1979). MR 82m:57003
  • 9. D. Gaboriau, G. Levitt, F. Paulin; Pseudogroups of isometries and Rips' theorem on free actions on $\mathbb {R} $-trees. Israel J. Math. 87 (1994), 403--428.
  • 10. D. Girault-Beauquier, M. Nivat; Tiling the plane with one tile. Topology and Category Theory from Computer Science, Oxford Univ. Press (1991) 291-334. CMP 92:06
  • 11. G. Levitt; Propriétés topologiques des $1$-formes fermées singulières. preprint.
  • 12. M. Keane; Non-ergodic interval exchange transformations, Israel J. Math 26, no. 2, (1977):188-196. MR 55:8313
  • 13. R. Kenyon; Tiling with squares and square-tilable surfaces. preprint, ENS-Lyon 1994.
  • 14. R. Kenyon; Rigidity of planar tilings, Invent. Math. 107 (1992):637-651 Erratum, Invent. Math. 112 (1993), p. 223. MR 92m:52049
  • 15. C. Kenyon, R. Kenyon; Tiling polygons with rectangles. Proc. 33rd FOCS, (1992):610-619.
  • 16. R. C. Lyndon, P. E. Shupp; Combinatorial group theory, Springer, 1977. MR 58:28182
  • 17. W. Magnus, A. Karrass, D. Solitar; Combinatorial Group Theory: presentations of groups in terms of generators and relations. 2nd ed., Dover publications, New York. MR 34:7617
  • 18. J. Morgan, P. Shalen; Free actions of surface groups on $\mathbf {% R}$-trees. Topology 30, (1991), 143-154. MR 92e:20016
  • 19. P. Shalen; Dendrology of groups: an introduction, in: Essays in group theory, S. Gersten, editor, MSRI publications, Springer, 1987. MR 89d:57012
  • 20. W. P. Thurston; Conway's tiling groups. Amer. Math. Monthly, 97 (1990):757-773. MR 91k:52028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20F34, 20E08, 58F03

Retrieve articles in all journals with MSC (1991): 20F34, 20E08, 58F03

Additional Information

Richard Kenyon
Affiliation: CNRS UMR 128, Ecole Normale Superieure de Lyon, 46, allée d’Italie, 69364 Lyon, France

Keywords: $\R$-tree, topological $1$-form, pseudo-Anosov diffeomorphism, tiling
Received by editor(s): June 30, 1994
Received by editor(s) in revised form: June 20, 1995
Additional Notes: This work was partially completed while the author was at the Institut Fourier, Grenoble
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society