Appears in collection : 2014 - T1 - WS1 - Asymptotic properties of groups

A finitely generated group has subexponential growth if the number of group elements expressible as words of length $\le n$ growssubexponentially in $n$. I will show that every countable group that does not contain asubgroup of exponential growth imbeds in a finitely generated group ofsubexponential growth. This shows that there are no restrictions on being a subgroup of a group of exponential growth, except the obvious ones. This produces in particular the first examples of groups ofsubexponential growth containing $\mathbb Q$. This also producesgroups of subexponential growth and arbitrarily large distortion in uniformly convex Banach (e. g. \ Hilbert) spaces. This is joint work with Anna Erschler.