On groups interpretable in various valued fields (Q6565769)

The authors continue their work from [\textit{Y. Halevi} et al., Adv. Math. 404, Part A, Article ID 108408, 58 p. (2022; Zbl 1505.03084)], where they studied fields interpretable in valued fields, and extend their investigation to interpretable groups. They analyze groups interpretable in the following three important families of valued fields:\N\begin{itemize}\N\item[1.] \(V\)-minimal (certain expansions of algebraically closed valued fields of residue characteristic 0),\N\item[2.] power bounded \(T\)-convex (certain expansions of real closed valued fields),\N\item[3.] \(p\)-adically closed fields (elementarily equivalent to finite extensions of \(\mathbb{Q}_p\)).\N\end{itemize}\NIt is shown (Theorem 1.1) that any such infinite group \(G\) is (up to the quotient by a finite normal subgroup) definably isomorphic to a group in one of four distinguished sorts: the underlying valued field, its residue field, its value group or its quotient by the value ring. This is an impressive reduction comparing to the usual ``geometric sorts'' for which the elimination of imaginaries is known.\N\NAs a corollary, it is shown (Theorem 1.3) that \(G\) has unbounded exponent and if \(G\) is dp-minimal then it is abelian-by-finite. Theorem 1.3 implies, in particular, that every 1-dimensional group definable in any of the above structures is abelian-by-finite generalizing results by \textit{A. Pillay} and \textit{N. Yao} [Arch. Math. Logic 58, No. 7--8, 1029--1034 (2019; Zbl 1468.03047)], and \textit{A. Onshuus} and \textit{M. Vicaría} [Ann. Pure Appl. Logic 171, No. 6, Article ID 102795, 27 p. (2020; Zbl 1481.03025), Theorem 1.1].