# Marcin Sabok: Automatic continuity for isometry groups

Place: Fields Institute, Room 210
Date and time: Friday 16 January 2015 (13:30-15:00)
Speaker:  Marcin Sabok
Title: Automatic continuity for isometry groups

Abstract: We present a general framework for automatic continuity  results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the  groups of isometries of the Urysohn space and the Urysohn  sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\mathrm{Aut}([0,1],\lambda)$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov. The results and proofs are stated in the language of model theory for metric structures.