A length function $\ell$ on a group $G$ is a function from $G$ to the nonnegative real numbers satisfying the following for all group elements $x$ and $y$: $\ell(x)=0$ if and only if $x=1_G$, $\ell(x^{-1}) = \ell(x)$, and $\ell(xy) \leq \ell(x)+\ell(y)$.
In this talk, we will investigate the asymptotic behavior of compatible length functions on Polish groups, and in particular, the extent to which a sphere of large radius with respect to one length function looks spherical with respect to another. This is joint work with Christian Rosendal. | 
