Automorphism groups of affine varieties consisting of algebraic elements

Abstract

Given an affine algebraic variety X X , we prove that if the neutral component A u t ∘ ( X ) \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G G contains a closed connected nested ind-subgroup H ⊂ G H\subset G , and for any g ∈ G g\in G some positive power of g g belongs to H H , then G = H G=H .