A descending chain condition for groups definable in -minimal structures

We prove that if G is a group definable in a saturated o -minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G 00 of bounded index and G / G 00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.