  We define and study the notion of \emph{ample metric generics} for a Polish
topological group, which is a weakening of the notion of ample generics
introduced by Kechris and Rosendal in \cite{Kechris-Rosendal:Turbulence}. Our
work is based on the concept of a \emph{Polish topometric group}, defined in
this article. Using Kechris and Rosendal's work as a guide, we explore
consequences of ample metric generics (or, more generally, ample generics for
Polish topometric groups). Then we provide examples of Polish groups with ample
metric generics, such as the isometry group $\Iso(\bU_1)$ of the bounded
Urysohn space, the unitary group ${\mathcal U}(\ell_2)$ of a separable Hilbert
space, and the automorphism group $\Aut([0,1],\lambda)$ of the Lebesgue measure
algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and
Tsankov that this last group has the automatic continuity property, i.e., any
morphism from $\Aut([0,1],\lambda)$ into a separable topological group is
continuous.
