ON NON-COMPACT <i>p</i>-ADIC DEFINABLE GROUPS

DEFINABLE GROUPS AND COMPACT p-ADIC LIE GROUPS

We formulate a version of the o-minimal group conjectures from [11], which is appropriate for groups G definable in a (saturated) p-adically closed field K, We discuss the conjectures in two cases, when G is defined over Qp and when G is of the form E(K) for E an elliptic curve over K.

O-minimal cohomology and definably compact definable groups

LetN an o-minimal expansion of a real closed field. We develop the cohomology theory for the category ofN -definable manifolds with continuous N -definable maps and use this to solve the Peterzil-Steinhorn problem [ps] on the existence of torsion points onN -definably compact N -definable abelian groups. Namely we prove the following result: Let G be an N -definably compact N -definably connect...

Compact domination for groups definable in linear o-minimal structures

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

Let M = 〈M, +, <, 0, S〉 be a linear o-minimal expansion of an ordered group, and G = 〈G,⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ∨ definable subgroup U of 〈Mn, +〉 and a lattice L of rank equal to the dimension of the ‘compact part’ of G.

On divisibility in definable groups

Let M be an o–minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. In this note, we show that a definably compact definably connected group is divisible.