## Asymptotic behavior and rectangular band structures in SL$$(2,\mathbb{R})$$.(English)Zbl 0982.22005

In this article the authors study in some detail subsemigroups of Sl$$(2,\mathbb R)$$ not contained in a Borel subgroup. They attach to each such subsemigroup an “asymptotic object”, an idempotent semigroup called a rectangular band defined on a closed subset of a toral surface; this idempotent semigroup plays a key role in their investigations. They also introduce “umbrellas”, limiting directions at infinity (in the Lie algebra) of a subsemigroup. A principal result is that for a connected open subsemigroup of Sl$$(2,\mathbb R)$$ the umbrella is convex, and is in fact the interior of a three-dimensional Lie semialgebra. Other applications include the classification of exponential subsemigroups and the asymptotic behavior of semigroups of integer matrices.

### MSC:

 22E15 General properties and structure of real Lie groups 22E46 Semisimple Lie groups and their representations 22A15 Structure of topological semigroups 22A25 Representations of general topological groups and semigroups
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