Semialgebraic groups have been intensively studied in the last three decades, and it is a field of current research. Some approaches to the study of low-dimensional semialgebraic groups, over the ordered field of the real numbers, have been offered by Razenj (1991), Strzebonski (1993), and Madden and Stanton (1992). So far there is no classification of definably compact definably connected groups definable over an o-minimal structure, up to definable group isomorphisms, as we do have for Lie groups. Even more, such classification does not exist for semialgebraic groups over a real closed field either.
In this talk I will present the results we have obtained in my doctoral thesis project on the classification of one-dimensional semialgebraically connected semialgebraic groups over a real closed field. For this, I will show some results on the existence of definable local homomorphisms between definably compact semialgebraic groups and algebraic groups with generic neighborhoods, as well as some results on covering maps in the category of locally definable groups in o-minimal expansions of real closed fields.
This PhD project is supervised by Alf Onshuus and Kobi Peterzil.
