In several papers, we studied interpretable groups and fields in certain expansions of a valued field \(K\). The analysis was based on associating to an interpretable group \(G\) a type-definable, “infinitesimal” subgroup \(\nu\) coming from four distinguished sorts: \(K\), its residue field, the value group, or \(K/\mathcal{O}\) (where \(\mathcal{O}\) is the valuation ring). When \(G\) was either the additive group of an interpretable field or definably semisimple, then only one of the first two sorts can appear in \(\nu\), and in those cases, \(\text{rank}(\nu) = \text{rank}(G)\).

In a recent paper, we showed that \(\nu\) breaks into a direct product of its four subgroups (coming from each of the sorts) and gave examples where \(\text{rank}(\nu) < \text{rank}(G)\).

In this talk I plan to describe the whole project and if time permits also the more recent results.

This is a joint work with Yatir Halevi and Assaf Hasson.