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Journal of Lie Theory 14 (2004), No. 1, 073--109
Copyright Heldermann Verlag 2004

On Compactification Lattices of Subsemigroups of SL(2, R)

Brigitte E. Breckner
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Str. M. Kogalniceanu 1, 3400 Cluj-Napoca, Romania,

Wolfgang A. F. Ruppert
Institut für Mathematik und Angewandte Statistik, Universität für Bodenkultur, Peter Jordanstr. 82, 1190 Wien, Austria,

Using the tools introduced in a previous article of the authors [ J. Lie Theory 11 (2001), 559--604] we investigate topological semigroup compactifications of closed connected submonoids with dense interior of Sl(2, R). In particular, we show that the growth of such a compactification is always contained in the minimal ideal, and describe the subspace of all minimal idempotents (typically a two-cell) and the maximal subgroups (these are always isomorphic to a compactification of R). For a large class of such semigroups we give explicit constructions yielding all possible topological semigroup compactifications and determine the structure of the compactification lattice.

Keywords: Bohr compactification, lattice of compactifications, asymptotic homomorphism, subsemigroups of Sl(2, R), Lie semigroups, Lie semialgebras, diamond product, rectangular domain, umbrella set, divisible semigroup, UDC semigroup.

MSC 2000: 22E15, 22E46, 22A15, 22A25.

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