A quasi-isometric embedding theorem for groups

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F{\o}lner functions, and elementary classes of amenable groups

Fundamental groups of asymptotic cones

We show that for any metric space $M$ satisfying certain natural conditions, there is a finitely generated group $G$, an ultrafilter $\omega$, and an isometric embedding $\iota$ of $M$ to the asymptotic cone ${\rm Cone}_\omega (G)$ such that the induced homomorphism $\iota ^ \ast :\pi_1(M)\to \pi_1({\rm Cone}_\omega (G))$ is injective. In particular, we prove that any countable group can be embedded into a fundamental group of an asymptotic cone of a finitely generated group.Comment: This is a corrected version of the paper. Some proofs are improved and several typos are corrected. The main result remains unchange

Acylindrically hyperbolic groups with exotic properties

We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group $Q$ with strong fixed point properties: $Q$ has property $FL^p$ for all $p\in [1, +\infty)$, and every action of $Q$ on a finite dimensional contractible topological space has a fixed point. In addition, $Q$ has other properties which are rather unusual for groups exhibiting "hyperbolic-like" behaviour. E.g., $Q$ is not uniformly non-amenable and has finite generating sets with arbitrary large balls consisting of torsion elements

Evolutionarity of MHD shock waves in collisionless plasma with heat fluxes

The evolutionarity conditions for the MHD shock waves are considered within the framework of the 8-moment approximation for collisionless plasma with heat fluxes. In the general case, evolutionarity diagrams are obtained depending on the relative magnitude of the Alfven wave velocity in front of or behind the shock wave front. The evolutionarity conditions for parallel shock waves are analyzed using previously obtained solutions for parallel MHD shock waves in collisionless plasma with heat fluxes. On the plane of dimensionless parameters characterizing plasma velocity and heat flux in front of the shock wave, the regions of evolutionarity are determined for the fast and slow shock waves propagating along the magnetic field.Comment: 22 pages, 11 figure

Synthetic Approaches to Novel Pyridine and Indole Derivatives as Potential Agents for the Treatment of Neurodegenerative Disorders

Alzheimer’s Disease (AD), Parkinson’s Disease (PD) and Lewy Body Disease (LBD) are some of the many neurodegenerative disorders associated with dementia, for which there is no ultimate cure. It is widely accepted that central nervous system (CNS) nicotinic acetylcholine receptors (nAChRs) may be strongly implicated in the pathology of these devastating disorders, and that stimulation of nAChRs can enhance cognitive behaviour in animals and humans. Nicotine and other nicotinic receptor binding compounds have, over many years, been explored as potential therapies for disorders such as AD and PD. This thesis describes the preparation and pharmacological investigation of a series of 3- substituted and 3,5-disubstitued pyridine derivatives as potential novel and selective nictotinic receptor agonists. Chapter Two details the synthesis of targeted compounds using the generation of [(pyridin-3-yl)methyl]lithium and [(5-methylpyridin-3- yl)methyl]lithium, respectively and subsequent reaction with various electrophiles. Unsuccessful attempts at the synthesis of enantiomerically pure 4-substituted arylpyridin-3-yl-ethanol derivatives by reduction of prochiral 4-substituted arylpyridin-3-yl-ethanone derivatives were made using both catalytic and enzymatic approaches; however, a pair of enantiomerically pure alcohols were isolated via the resolution of diastereomeric esters (prepared by reaction with (S)-O-acetyl mandelic acid) and subsequent hydrolysis. iv Chapter Three explores the synthesis of targeted compounds using halogen-lithium exchange reactions of 3-bromopyridine using n-BuLi and ring-opening by the resultant pyridin-3-yllithium of 4-substituted aryl epoxides. As an extension, Sonogashira cross-coupling of 3- bromopyridine and 4-substituted arylacetylenes and subsequent hydration as an approach to 4-substituted pyridin-3-yl-ethanone derivatives is described. A series of indole derivatives were synthesised using identical approaches. Using methodology developed in previous Chapters, Chapter Four describes approaches to symmetrical and asymmetrical 3,5- bis(arylethynyl)pyridine derivatives, the corresponding bis(ketones), alcohols and 3,5-disubstituted keto-alcohol products. Chapter Five details preliminary pharmacological data (binding and functional assays) performed by our collaborators at Institut de Recherches Servier

Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is proved for quasi--geodesics instead of geodesics. This allows to simplify the exposition in the last section. To appear in Invent. Mat