A geometric generalization of Kaplansky’s direct finiteness conjecture

Let G G be a group and let alttext="k"> k encoding="application/x-tex">k field. Kaplansky’s direct finiteness conjecture states that every one-sided unit of the ring alttext="k left-bracket upper G right-bracket"> [ stretchy="false">] encoding="application/x-tex">k[G] must two-sided unit. In this paper, we establish geometric theorem for endomorphisms symbolic algebraic varieties. Whenever is sofic or more generally surjunctive group, our result implies generalization near R left-parenthesis k comma right-parenthesis"> R stretchy="false">( , stretchy="false">) encoding="application/x-tex">R(k,G) which X Subscript g Baseline colon element-of X g : ∈<!-- ∈ encoding="application/x-tex">k[X_g\colon \in G] as contains naturally subring homogeneous polynomials degree one. We also prove stable consequence Gottschalk’s Surjunctivity Conjecture.