Finitely approximable groups and actions Part II: Generic representations

Given a finitely generated group $\Gamma$, we study the space ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ of all actions of $\Gamma$ by isometries of the rational Urysohn metric space ${\mathbb Q\mathbb U}$, where ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ is equipped with the topology it inherits seen as a closed subset of ${\rm Isom}({\mathbb Q\mathbb U})^\Gamma$. When $\Gamma$ is the free group \F_n on $n$ generators this space is just ${\rm Isom}({\mathbb Q\mathbb U})^n$, but is in general significantly more complicated. We prove that when $\Gamma$ is finitely generated Abelian there is a generic point in ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$, i.e., there is a comeagre set of mutually conjugate isometric actions of $\Gamma$ on ${\mathbb Q\mathbb U}$

Infinite asymptotic games

We study infinite asymptotic games in Banach spaces with an F.D.D. and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into $\ell_p$ sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. These results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences

Incomparable, non isomorphic and minimal Banach spaces

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence