Automorphism groups of affine varieties without non-algebraic elements

Given an affine algebraic variety X, we prove that if the neutral component 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 contains a closed connected nested ind-subgroup H ⊂ G, and for any g ∈ G some positive power of g belongs to H, then G = H.