Definably simple groups in o-minimal structures

Abstract:

Let $\mathbb {G}=\langle G, \cdot \rangle$ be a group definable in an o-minimal structure $\mathcal {M}$. A subset $H$ of $G$ is $\mathbb {G}$-definable if $H$ is definable in the structure $\langle G,\cdot \rangle$ (while definable means definable in the structure $\mathcal {M}$). Assume $\mathbb {G}$ has no $\mathbb {G}$-definable proper subgroup of finite index. In this paper we prove that if $\mathbb {G}$ has no nontrivial abelian normal subgroup, then $\mathbb {G}$ is the direct product of $\mathbb {G}$-definable subgroups $H_1,\ldots ,H_k$ such that each $H_i$ 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.