Groups definable in linear o-minimal structures: the non-compact case

Connected components of definable groups and o-minimality I

We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G00 6= G000, yielding also new examples of “non G-compact” theories. We also prove that for G definable in a (saturated) o-minimal structure, G has a “bounded orbit” (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably amenable, giving a positive answer to a conje...

Lie-like Decompositions of Groups Definable in O-minimal Structures

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N ; the quotient G/N always has maximal definably compact ...

Groups, measures, and the NIP

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups G in saturated o-minimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant...

Compact domination for groups definable in linear o-minimal structures

Topology of Definable Abelian Groups in O-minimal Structures

In this note we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G) 6= 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.