Groups definable in linear o-minimal structures

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Abstract

Let M = <M, <, +, 0, …> be a linear o-minimal expansion of an ordered group, and G 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 V-definable subgroup U of <Mn, +> and a lattice L of rank equal to the dimension of the ‘compact part’ of G. This is suggested as a structure theorem analogous to the classical theorem that every connected abelian Lie group is Lie isomorphic to a direct sum of copies of the additive group <R, +> of the reals and the circle topological group S1. We then apply our analysis and prove Pillay’s Conjecture andthe Compact Domination Conjecture for a saturated M as above. En route, we show that the o-minimal fundamental group of G is isomorphic to L. Finally, we state some restrictions on L.

Attributes

Attribute NameValues
URN
  • etd-07202007-144313

Author Pantelis E. Eleftheriou
Advisor Gregory Madey
Contributor Sergei Starchenko, Committee Member
Contributor Julia Knight, Committee Member
Contributor Lou van den Dries, Committee Member
Contributor Steven Buechler, Committee Member
Contributor Gregory Madey, Committee Chair
Degree Level 2
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Defense Date
  • 2007-06-29

Submission Date 2007-07-20
Country
  • United States of America

Subject
  • groups

  • o-minimal structures

Publisher
  • University of Notre Dame

Language
  • English

Access Rights Open Access
Content License
  • All rights reserved

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