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.


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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