Normalisers in Limit Groups

Let $\text{G}$ be a limit group, $S\subset \text{G}$ a subgroup, and $N$ the normaliser of $S$. If $H_1(S,\mathbb{Q})$ has finite $\text{Q}$-dimension, then $S$ is finitely generated and either $N\mathrm{/}S$ is finite or $N$ is abelian. This result has applications to the study of subdirect products of limit groups.