### On maximal symmetry in Banach spaces and isometric representations of Polish groups.

It follows from the theory of Lipschitz free Banach spaces that
any Polish group is topologically isomorphic to an index 2 closed subgroup
of the isometry group of a separable Banach space. However, this result
gives almost no information about the relation between the isomorphic
structure of the group and the Banach space.
As a reversal, we study which groups can appear as the isometry group of a
fixed separable Banach space
under equivalent renormings. In particular, we answer a longstanding problem
of G. Wood on whether GL(X) always contains a maximal bounded subgroup. This
is joint work with V. Ferenczi.

*Presentation*