Anti-affine algebraic groups

We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.Comment: Prior work of Carlos Sancho de Salas acknowledged, additional minor changes

Affine algebraic groups with periodic components

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.Comment: 20 page

Some basic results on actions of non-affine algebraic groups

We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups.Comment: 20 pages ; references and final example adde

Epimorphic subgroups of algebraic groups

In this note, we show that the epimorphic subgroups of an algebraic group are exactly the pull-backs of the epimorphic subgroups of its affinization. We also obtain epimorphicity criteria for subgroups of affine algebraic groups, which generalize a result of Bien and Borel. Moreover, we extend the affinization theorem for algebraic groups to homogeneous spaces.Comment: Final version, accepted for publication at Mathematical Research Letter