**
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

Romania

brigitte@math.ubbcluj.ro

and

Institut für Mathematik und

Angewandte Statistik

Universität für Bodenkultur

Peter Jordanstr. 82

A-1190 Wien

Austria

breckner@edv1.boku.ac.at

ruppert@edv1.boku.ac.at

**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

**Full text of the article:**

[Previous Article] [Contents of this Number]

*
© 2001 ELibM for
the EMIS Electronic Edition
*