Automorphism groups of affine varieties consisting of algebraic elements

Given an affine algebraic variety X X, we prove that if the neutral component A u t ∘ (X) Aut^ (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 G, and for any g ∈ G g G some positive power of g g belongs to H H, then G = H G=H.