Journal of Lie Theory, Vol. 11, No. 2, pp. 559-604 (2001)

On Asymptotic Behavior and Rectangular Band Structures in SL(2, R)

Brigitte E. Breckner and Wolfgang A.F. Ruppert

Babes-Bolyai University
Faculty of Mathematics and
Computer Science
Str. M. Kogalniceanu 1
RO-3400 Cluj-Napoca
Institut für Mathematik und
Angewandte Statistik
Universität für Bodenkultur
Peter Jordanstr. 82
A-1190 Wien

Abstract: We associate with every subsemigroup of $Sl(2, R)$, not contained in a single Borel group, an `asymptotic object,' a rectangular band which is defined on a closed subset of a torus surface. Using this concept we show that the horizon set (in the sense of Lawson) of a connected open subsemigroup of $Sl(2, R)$ is always convex, in fact the interior of a three dimensional Lie semialgebra. Other applications include the classification of all exponential subsemigroups of $Sl(2, R)$ and the asymptotics of semigroups of integer matrices in $Sl(2, R)$.

Keywords: asymptotic objects, asymptotic property, subsemigroups of $Sl(2, R)$, Lie semigroups, Lie semialgebras and their classification, compression semigroups, diamond product, rectangular domain, umbrella sets, Control Theory in Lie groups, asymptotics of integer matrix semigroups, semialgebraic sets

Classification (MSC2000): 22E15, 22E67, 22E46, 22A15, 22A25

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