Weak amenability of CAT(0)-cubical groups

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations π _z for which the quantities z ^{ℓ (g)} are matrix coefficients. Here ℓ is a length function on G obtained from the combinatorial distance function on the complex X.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

Enzymatic determination of succinic acid in foodstuffs

Ohne Zusammenfassung

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Enzymatic determination of succinic acid in foodstuffs

     

Constructions preserving Hilbert space uniform embeddability of discrete groups

Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

     

Exactness and Uniform Embeddability of Discrete Groups

A numerical quasi-isometry invariant R() of a finitely generatedgroup is defined whose values parametrize the difference between being uniformly embeddable in a Hilbert space and () being exact.     

Photocontrol of Polar Aromatic Interactions by a Bis‐Hemithioindigo Based Helical Receptor

The first example of a bis‐hemithioindigo (bis‐HTI)‐based molecular receptor was realized. Its folding and selective binding affinity for aromatic guest molecules can be precisely controlled by visible light and heat. The thermodynamically stable state of the bis‐HTI is the s‐shaped planar Z,Z‐configuration. After irradiation with 420 nm light only the E,Z‐configuration is formed in a highly selective photoisomerization. The E,Z‐isomer adopts a helical conformation because of the implementation of repulsive steric interactions. The E,Z‐configured helix is able to recognize electron‐poor aromatic guests exclusively through polar aromatic interactions and also distinguishes between regioisomers. After heating, the Z,Z‐configuration is completely restored and the aromatic guest molecule is efficiently released.     

Tuning the Ground and Excited State Dynamics of Hemithioindigo Molecular Motors by Changing Substituents

Efficiency and performance of light triggered molecular motors are crucial features that need to be mechanistically understood to improve the performance and enable conscious property tailoring for specific applications. In this work, three different hemithioindigo-based molecular motors are investigated and all four steps in their complete unidirectional rotation are unraveled fully quantitatively. Transient absorption spectroscopy across twelve orders of magnitude in time is used to probe the fs nuclear motions up to the ms thermal kinetics, covering the timeframe of the whole motor rotation. The newly known full mechanisms allow simulation of the motor systems to scrutinize their performance at realistic illumination conditions. This highlights the importance of photoisomerization quantum yields for the rotation speed. The substitution pattern in close proximity to the rotation axle influences the excited and ground state properties. Reduction of electron donation and concomitant increase of steric hindrance leads to faster photoisomerization reactions with quasi-ballistic behavior, but also to a slight decrease in the quantum efficiency. The expected decelerating effects of increased sterics are primarily manifested in the ground state. A promising approach for next-generation hemithioindigo motors is to elevate electron donation at the rotor fragment followed by an increase of steric hindrance.     

Coarse Non-Amenability and Coarse Embeddings

We construct the first example of a coarsely non-amenable (= without Guoliang Yu’s property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.     

Uniform embeddings of bounded geometry spaces into reflexive Banach space

We show that every metric space with bounded geometry uniformly embeds into a direct sum of spaces ('s going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed space. In the case of discrete groups we prove the analogue of a--menability - the existence of a metrically proper affine isometric action on a direct sum of spaces.