ON SYMBOLIC ALGEBRAIC GROUP VARIETIES AND DUAL SURJUNCTIVITY

The candidate solution accurately restates and follows the same core arguments and structural lemmas as the paper: (a) for sofic groups over an uncountable algebraically closed field, post-surjectivity implies (•)-pre-injectivity, proved via a uniform post-surjectivity lemma and sofic graph approximations leading to a counting/dimension contradiction; and (b) for amenable groups (any algebraically closed field), pre-injectivity implies surjectivity (Myhill), and surjectivity implies both (•)- and (••)-pre-injectivity (Moore), via algebraic mean dimension and tiling. Minor differences are cosmetic (e.g., the candidate’s “fiber-dimension” phrasing vs. the paper’s explicit connected-component counting), and the candidate omits the paper’s global convention that G is finitely generated. Substantively, both are aligned.