Scott showed that for every countable structure $A$, there is an $L_{\omega_1,\omega}$ sentence, called the Scott sentence, whose countable models are isomorphic copies of $A$. The quantifier complexity of a Scott sentence can be thought of as a measure of the complexity of a "description" of the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely-generated structure always have a $\Sigma_3$ Scott sentence. In this talk, we will focus on finitely-generated groups. On the one hand, most "natural" finitely-generated groups have a d-$\Sigma_2$ Scott sentence. On the other hand, we give a characterization of finitely generated structures where the $\Sigma_3$ Scott sentence is optimal. As an application, we give a construction of a finitely generated group where the $\Sigma_3$ Scott sentence is optimal.

This is joint work with Matthew Harrison-Trainor.