Covering a Polish group by translates of a nowhere dense set

We show that, for every nonlocally compact Polish group G with a left-invariant complete metric ρ, we have cov G = cov ( M ) . Here, cov G is the minimal number of translates of a fixed closed nowhere dense subset of G , which is needed to cover G , and cov ( M ) is the minimal cardinality of a cover of the real line R by meagre sets.