Rechtwinkelige Gebäude und lokalkompakte Gruppen
Zusammenfassung der Projektergebnisse
The project studied abstract group homomorphisms f : G −→ U from a sufficiently nice topological group G to a locally compact group U. The group U is a universal group in the sense of Burger-Mozes and De Medts-da Silva, acting on a right-angled building. The aim was to show that under certain geometric assumptions f is continuous and open. One main result of the project is as follows. Theorem C (Beßmann [Be1]) Let ∆ be a right-angled building of type (W, I). Consider the following conditions (i), (ii) or (iii). (i) If ∆ is irreducible and if there exists a spherical residue of type i ∪ i⊥ for some i ∈ I, then assume that the local group for i acts 2-transitively on the set of i-colours. (ii) If ∆ is irreducible and there exists no spherical residue of type i∪i⊥ for i ∈ I, then let i, j ∈ I such that mi,j = ∞ and assume that the local groups for i and j act 2-transitively on the set of i- and j-colours, respectively. Assume further that for each k ∈ i⊥ ∪ j ⊥ the local group for k acts 2-transitively on the set of k-colours. (iii) If ∆ is reducible, assume (ii) for each component. Let U be a universal group associated to ∆ satisfying one of the assumptions (i), (ii) or (iii). Let G be a Polish group and assume that φ : G → U is an abstract group homomorphism such that ker(φ) is an analytic subset of G. If φ is surjective, then φ is continuous and open. Beßmann’s Thesis [Be1] contains many more results on such homomorphisms. Another main result is as follows. Theorem A (Beßmann [Be1, Be2]) There exist topologically isomorphic universal groups associated to different right-angled buildings of the same type with different thicknesses.
