### A Characterization of SIN-Groups.

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Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $\mu \u0332\left(AB\right)\ge min\left(\mu \u0332\right(A)+\mu \u0332(B),\mu (G\left)\right)$ for unimodular G.

Let $N$ and $K$ be groups and let $G$ be an extension of $N$ by $K$. Given a property $\mathcal{P}$ of group compactifications, one can ask whether there exist compactifications ${N}^{\text{'}}$ and ${K}^{\text{'}}$ of $N$ and $K$ such that the universal $\mathcal{P}$-compactification of $G$ is canonically isomorphic to an extension of ${N}^{\text{'}}$ by ${K}^{\text{'}}$. We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties $\mathcal{P}$ and then apply this result to the almost periodic and weakly almost periodic compactifications of $G$.

We give a complete characterization of the locally compact groups that are non elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semiregular trees acting doubly...

We study the relationship between the classical invariance properties of amenable locally compact groups G and the approximate diagonals possessed by their associated group algebras L¹(G). From the existence of a weak form of approximate diagonal for L¹(G) we provide a direct proof that G is amenable. Conversely, we give a formula for constructing a strong form of approximate diagonal for any amenable locally compact group. In particular we have a new proof of Johnson's Theorem: A locally compact...

We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.