Locally definable subgroups of semialgebraic groups

Abstract

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let GG be an abelian semialgebraic group over a real closed field RR and let XX be a semialgebraic subset of GG. Then the group generated by XX contains a generic set and, if connected, it is divisible. More generally, the same result holds when XX is definable in any o-minimal expansion of RR which is elementarily equivalent to Ran,expℝan,exp. We observe that the above statement is equivalent to saying: there exists an mm such that Σmi=1(X−X)Σi=1m(X−X) is an approximate subgroup of GG.