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Transactions of the American Mathematical Society

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



Piecewise $ {\rm SL}\sb 2{\bf Z}$ geometry

Author: Peter Greenberg
Journal: Trans. Amer. Math. Soc. 335 (1993), 705-720
MSC: Primary 52B20
MathSciNet review: 1140914
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Abstract: Piecewise $ {\text{SL}}_2{\mathbf{Z}}$ geometry studies properties of the plane invariant under $ {\text{pl}}$-homeomorphisms which, locally, have the form $ x \mapsto Ax + b$ , with $ A \in {\text{SL}}_2{\mathbf{Z}}$, $ b \in {{\mathbf{Q}}^2}$ , and whose singular lines are rational. In this paper, invariants of polygons are obtained, relations with Pick's theorem are described, and a conjecture is posed.

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  • [D] R. L. Devaney, A piecewise linear model for the zones of instability of an area preserving map, Phys. D 10 (1984), 387-393. MR 763480 (86d:58031)
  • [GL] L. C. Glaser, Geometrical combinatorial topology, Vol. I, Van Nostrand Reinhold, New York, 1970.
  • [G] M. C. Golumbic, Special issue on interval graphs and related topics, Discrete Math. 55 (1985), 113-123. MR 798527 (86k:05095)
  • [HT] A. Hatcher and W. Thurston, A presentation for the mapping class group of an oriented surface, Topology 19 (1980), 221-237. MR 579573 (81k:57008)
  • [I] N. V. Ivanov, Complexes of curves and Teichmüller modular groups, Uspekhi Mat. Nauk 42 (1987), 49-91. MR 896878 (88g:32040)
  • [R] H. Rademacher, Topics in analytic number theory, Springer, Berlin, 1973. MR 0364103 (51:358)
  • [GS] P. Greenberg and V. Sergiescu, An acyclic extension of the braid group, preprint. MR 1090167 (92b:57004)
  • [Gr] P. Greenberg, A combinatorial method for pseudogroups, preprint.
  • [GKW] R. W. Gaskell, M. S. Klamkin, and P. Watson, Triangulations and Pick's theorem, Math. Mag. 49 (1976), 35-37. MR 0400046 (53:3881)

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Keywords: Piecewise linear homeomorphisms
Article copyright: © Copyright 1993 American Mathematical Society

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