We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G 00 = G 000 , 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 conjecture of Newelski and Petrykowski in this special case of groups definable in ominimal structures. We also introduce and discuss further conjectures on bounded orbits in definable groups. These results and analyses are informed by a decomposition theorem for groups in o-minimal structures.
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field. 2010 Mathematics Subject Classification. 03C64; 20J06; 06F25; 22E25.
A characterization of groups definable io o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0subgroups. Along the way we give structural theorems for solvable groups, linear groups, . and extensions of definably compact by torsion-free definable groups.
We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R · S where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.
Let
${\cal N}\left( G \right)$
be the maximal normal definable torsion-free subgroup of a
group G definable in an o-minimal structure M.
We prove that the quotient
$G/{\cal N}\left( G \right)$
has a maximal definably compact subgroup K,
which is definably connected and unique up to conjugation. Moreover, we show
that K has a definable torsion-free complement, i.e., there is
a definable torsion-free subgroup H such that
$G/{\cal N}\left( G \right) = K \cdot H$
and
$K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$
. It follows that G is definably homeomorphic
to
$K \times {M^s}$
(with
$s = {\rm{dim}}\,G - {\rm{dim}}\,K$
), and homotopy equivalent to K. This gives a
(definably) topological reduction to the compact case, in analogy with Lie
groups.
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