Cartan subgroups of groups definable in o-minimal structures (Q2921052)

A subgroup \(Q\) of a group \(G\) is a \textit{Cartan subgroup} if \(Q\) is a maximal nilpotent subgroup of \(G\) and every finite index normal subgroup of \(Q\) is of finite index in its normalizer in \(G\). The present paper studies Cartan subgroups of groups definable in o-minimal structures, i.e. first-order structures \(\mathcal{M}=\langle M,<,\ldots\rangle\) such that \(<\) is a dense linear order without endpoints on \(M\) and every subset of \(M\) that is definable in \(\mathcal{M}\) is the union of a finite set and finitely many intervals with endpoints in \(M\cup\{\pm\infty\}\). The main result of this paper shows that, given a group \(G\) definable in an o-minimal structure \(\mathcal{M}\), Cartan subgroups of \(G\) exist, they are definable in \(\mathcal{M}\) and they fall into finitely many conjugacy classes. The paper also contains a stronger result for definably connected groups definable in o-minimal structures, where a group \(G\) definable in a structure \(\mathcal{M}\) is called \textit{definably connected} if \(G\) has no proper subgroup of finite index that is definable in \(\mathcal{M}\).