Groups definable in ordered vector spaces over ordered division rings

Abstract

Abstract. Let M = M, +, <, 0, {λ} λ∈D be an ordered vector space over an ordered division ring D, and G = G, ⊕, e G an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex -definable subgroup U of M n , + and a lattice L of rank n. As two consequences, we derive Pillay's conjecture for M as above and we show that the o-minimal fundamental group of G is isomorphic to L.

The universal covering homomorphism in o‐minimal expansions of groups

“…The main result of the paper [5], in the language of the theory of locally definable covering homomorphisms, is the following (compare with [5] Remark 6.14). For a related result see also [8].…”

“…On the other hand, in the paper [5], the second author and S. Starchenko use definable t-continuous paths to define the o-minimal fundamental group π 1 (G) of a definably t-connected, definable group G following the classical case in [7] and the case in o-minimal expansions of fields treated by Berarducci and Otero in [1]. We will adapt that definition to the category of locally definable groups.…”

“…We will adapt that definition to the category of locally definable groups. As in [5] we will run the definition in parallel with respect to the τ -topology of a definably connected locally definable group G and the usual topology on an arbitrary definable subset X of R n .…”

“…When G is a definably compact, abelian definable group, we conjecture that l above is the dimension of G. This is known to be the case when R is linear ( [5]) or R is an o-minimal expansion of a real closed field ( [4]). So the conjecture is open for R eventually linear but not linear.…”

Definable group extensions in semi‐bounded o‐minimal structures

“…See, for example, [ElSt,Lemma 3.7]. An assumption that R is linear is also existent there, but the proof uses only the fact that R has no poles.…”