{#sec_prelim}
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Marked groups and limit groups {#sec_prelim_limit}
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Sela introduced limit groups in [@Sela_diophantine1]. For background about Sela’s limit groups, see also [@CG_compactifying] or [@Pau_theorie].

A *marked group* $(G,S)$ is a finitely generated group $G$ together with a finite ordered generating family $S=(s_1,\dots,s_n)$. Note that repetitions may occur in $S$, and some generators $s_i$ may be the trivial element of $G$. Consider two groups $G$ and $G'$ together with some markings of the same cardinality $S=(s_1,\dots,s_n)$ and $S'=(s'_1,\dots,s'_n)$. A *morphism of marked groups* $h\co (G,S){\rightarrow}(G',S')$ is a homomorphism $h\co G{\rightarrow}G'$ sending $s_i$ on $s'_i$ for all $i\in\{1,\dots,n\}$. Note that there is at most one morphism between two marked groups, and that all morphisms are epimorphisms.

A *relation* in $(G,S)$ is an element of the kernel of the natural morphism $F_n{\rightarrow}G$ sending $a_i$ to $s_i$ where $F_n$ is the free group with basis $(a_1,\dots,a_n)$. Note that two marked group are isomorphic if and only if they have the same set of relations.

Given any fixed $n$, we define ${{\mathcal {G}}}_n$ to be the set of isomorphism classes of marked groups. It is naturally endowed with the topology such that the sets $N_R(G,S)$ defined below form a neighbourhood basis of $(G,S)$. 