Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Symmetries of BLT-Sets

We do the tentative beginnings of a study of BLT-sets of generalised quadrangles via their symmetries. In particular, the study of whorls about a line leads us to hyperbolic reflections preserving a BLT-set of Q(4, q).     

Minimally generated Boolean algebras

We introduce the class of minimally generated Boolean algebras, i.e. those algebras representable as the union of a continuous well-ordered chain of subalgebras A ₁ where A _i+1 is a minimal extension of A _i. Minimally generated algebras are closely related to interval algebras and superatomic algebras.     

Synthesis of a biotinated amphiphile

The synthesis of a biotinated amphiphile assembled from d-(+)-biotin, ethylene diamine as spacer, galactaric acid and 1-dodecylamine was achieved in six steps. The key step was the synthesis of a bisacetonide protected galactaric ester, the structure of which was determined by X-ray analysis. Aminolysis, spacer attachment, coupling with biotin and deprotection led to the amphiphilic galactaramide.     

Reactivity sequences of dipolarophiles towards diazocarbonyl compounds - MO perturbation treatment

Whereas diazomethane cycloadditions are only accelerated by electron-attracting substituents in the olefinic or acetylenic dipolarophile, the cycloadditions of diazoacetic, diazomalonic and diazo(phenylsulfonyl)acetic ester show in accordance with the PMO treatment U-shaped activity functions when log k₂ is plotted versus the lowest IP of the dipolarophiles.     

Photochemistry of 2,2-Dimethyl-2H-Furo[3,4-b]Pyran-4,7(3H, 5H)-Dione

Irradiation (λ > 340 nm) of the title compound 2a in t-BuOH affords 3,3,10,10-tetramethyl-2,8-dioxatricyclo-[4.3.2.0^1,6]undecane-5,9-dione (3) via photoextrusion of 2-methylpropene and subsequent photocycloaddition of 2a to the alkene. The same regioisomer 3 is formed selectively and in much higher yields on irradiating 2a in the presence of excess 2-methylpropene. Irradiation of 3 (λ = 300 nm) in t-BuOH in the presence of the same alkene gives a 6:1 mixture of spirooxetanes 7a and 7b but not α-cleavage products. In 2-propanol, 2a is photoreduced to a 2:1 mixture of diastereoisomeric hydrodimers 9 .