Definably simple groups in o-minimal structures

Abstract

Let G = ⟨ G , ⋅ ⟩ \mathbb {G}=\langle G, \cdot \rangle be a group definable in an o-minimal structure M \mathcal {M} . A subset H H of G G is G \mathbb {G} -definable if H H is definable in the structure ⟨ G , ⋅ ⟩ \langle G,\cdot \rangle (while definable means definable in the structure M \mathcal {M} ). Assume G \mathbb {G} has no G \mathbb {G} -definable proper subgroup of finite index. In this paper we prove that if G \mathbb {G} has no nontrivial abelian normal subgroup, then G \mathbb {G} is the direct product of G \mathbb {G} -definable subgroups H 1 , … , H k H_1,\ldots ,H_k such that each H i H_i is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.