Large scale geometry of Banach-Lie groups

We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of $C^*$-algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital $C^*$-algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-localy compact groups having the Haagerup property, most of them being non-amenable. These are the groups $\mathcal{U}_2(M,\tau)$, where $M$ is a semifinite von Neumann algebra with a normal faithful semifinite trace $\tau$. Finally, we investigate the groups $\mathrm{E}_n(A)$, which are closed subgroups of $\mathrm{GL}(n,A)$ generated by elementary matrices, where $A$ is a unital Banach algebra. We show that for $n\geq 3$, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if $A$ is an infinite-dimensional unital $C^*$-algebra, then $\mathrm{E}_2(A)$ does not have the Haagerup property. If $A$ is moreover abelian and separable, then $\mathrm{SL}(2,A)$ does not have the Haagerup property.