High transitivity in algebra and geometry

An infinite group G is called highly transitive if it acts on some infiniteset m-transitively for any natural number m. We give a brief survey on some recent results on abstract highly transitive groups.

Then we pass to examples of affine algebraic varieties with the automorphism group acting highly transitively; specifically, of toric affine varieties. We show that a highly transitive group can be generated by a finite number of one-parameter subgroups; for the affine spaces, three such subgroups suffice. We formulate some open problems related to group growth.