On the algebraic structure of the unitary group

We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length. A property of uncountable groups that has recently been studied by a number of authors is a strengthening of the property of uncountable cofinality that originated in the work of Jean-Pierre Serre on actions of groups on trees [7]. Here an uncountable group G is said to have uncountable cofinality if G is not the union of a countable increasing chain of proper subgroups. Serre proved this to be one of the three conditions in his reformulation of when a group does not have fixed point free actions without inversions on trees and it has subsequently been confirmed for a great number of profinite groups (see, e.g., Koppelberg and Tits [4]) and groups of permutations of N. The strengthening of this property, in which we are interested, comes from considering the additional condition on G that whenever E is a symmetric generating set for G