A geometric generalization of Kaplansky’s direct finiteness conjecture

Abstract

Let G G be a group and let k k be a field. Kaplansky’s direct finiteness conjecture states that every one-sided unit of the group ring k [ G ] k[G] must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever G G is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky’s direct finiteness conjecture for the near ring R ( k , G ) R(k,G) which is k [ X g : g ∈ G ] k[X_g\colon g \in G] as a group and which contains naturally k [ G ] k[G] as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky’s stable finiteness conjecture is a consequence of Gottschalk’s Surjunctivity Conjecture.