A geometric generalization of Kaplansky’s direct finiteness conjecture

Phung, Xuan Kien

doi:10.1090/proc/16333

A geometric generalization of Kaplansky’s direct finiteness conjecture
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by Xuan Kien Phung

Proc. Amer. Math. Soc. 151 (2023), 2863-2871

DOI: https://doi.org/10.1090/proc/16333

Published electronically: March 31, 2023

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Abstract:

Let $G$ be a group and let $k$ be a field. Kaplansky’s direct finiteness conjecture states that every one-sided unit of the group ring $k[G]$ must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever $G$ 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 $R(k,G)$ which is $k[X_g\colon g \in G]$ as a group and which contains naturally $k[G]$ 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.

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