$2$;

there are finitely many conjugacy classes of non-cyclic maximal abelian subgroups in $\Gamma$, and abelian subgroups of $\Gamma$ are finitely generated. More precisely, one has the following bound on the ranks of maximal abelian subgroups: $$\sum_A (\operatorname{Rk}A-1)\leq b_1(\Gamma)-1$$ where the sum is taken over the set of conjugacy classes of non-cyclic maximal abelian subgroups of $\Gamma$, and where $b_1(\Gamma)$ denotes the first Betti number of $\Gamma$;

$\Gamma$ has a finite classifying space, and the cohomological dimension of $\Gamma$ is at most $\max(2,r)$ where $r$ is the maximal rank of an abelian subgroup of $\Gamma$.

A combination theorem by Dahmani also shows that $\Gamma$ is hyperbolic relative to its non-cyclic abelian subgroups [@Dahmani_combination].

[[@Sela_diophantine1; @KhMy_irreducible1; @KhMy_irreducible2; @Pfander_finitely]]{}A limit group is finitely presented, its abelian subgroups are finitely generated, it has only finitely many conjugacy classes of maximal non-cyclic abelian subgroups, and it has a finite classifying space.

Finally, we can also easily derive from the dévissage theorem the existence of a *principal* splitting, a major step in Sela’s proof of the finite presentation of limit groups (see corollary \[cor\_principal\] and [@Sela_diophantine1 Theorem 3.2]).

Unlike Sela’s proof, the proof we give doesn’t need any JSJ theory, and does not use the 