Piecewise 𝑆𝐿₂𝑍 geometry

Greenberg, Peter

doi:10.1090/S0002-9947-1993-1140914-1

Piecewise $\textrm {SL}_ 2\textbf {Z}$ geometry
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by Peter Greenberg

Trans. Amer. Math. Soc. 335 (1993), 705-720

DOI: https://doi.org/10.1090/S0002-9947-1993-1140914-1

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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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