More precisely, if $n$ is a fixed integer, a marked group is a group together with an ordered generating family $S=(s_1,\dots,s_n)$. Two marked groups $(\Gamma,S)$ and $(\Gamma',S')$ are close to each other in this topology if for some large $R$, $(\Gamma,S)$ and $(\Gamma',S')$ have exactly the same relations of length at most $R$ (see section \[sec\_prelim\_limit\]).

Limit groups have several equivalent characterisations: a finitely generated group $G$ is a limit group if and only if it is fully residually free, if and only if it has the same universal theory as a free group, if and only if it is a subgroup of a non-standard free group [@Remeslennikov_exist_siberian; @CG_compactifying].

One of the main results about limit groups is a structure theorem due to Kharlampovich–Myasnikov, Pfander and Sela [@KhMy_irreducible1; @KhMy_irreducible2; @Pfander_finitely; @Sela_diophantine1]. This theorem claims that a limit group can be inductively obtained from free abelian groups and surface groups by taking free products and amalgamations over ${{\mathbb{Z}}}$ (see Theorem \[devissage\_simple\] below). This structure theorem implies that a limit group is finitely presented, and that its abelian subgroups are finitely generated. The goal of the paper is to give a simpler proof of this result in the broader context of groups 