  We prove several theorems relating amenability of groups in various
categories (discrete, definable, topological, automorphism group) to
model-theoretic invariants (quotients by connected components, Lascar Galois
group, G-compactness, ...). For example, if $M$ is a countable,
$\omega$-categorical structure and $Aut(M)$ is amenable, as a topological
group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is
compact, Hausdorff (also over any finite set of parameters), that is $T$ is
G-compact. An essentially special case is that if $Aut(M)$ is extremely
amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory
$T$ can be recovered from its category $Mod(T)$ of models. On the side of
definable groups, we prove for example that if $G$ is definable in a model $M$,
and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$
and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an
earlier paper of the authors.
  We also take the opportunity to further develop the model-theoretic approach
to topological dynamics, obtaining for example some new invariants for
topological groups, as well as allowing a uniform approach to the theorems
above and the various categories.
