All groups are surjunctive

TL;DR

This paper proves the Gottschalk surjunctivity conjecture for finitely generated groups, showing that their space of colorings cannot be strictly self-embedded, and also confirms Kaplansky's direct finiteness conjecture.

Contribution

It establishes the surjunctivity property for all finitely generated groups, resolving two longstanding conjectures in group theory.

Findings

Finitely generated groups are surjunctive.

The space of colorings admits no strict self-embedding.

Confirmed Kaplansky's direct finiteness conjecture.

Abstract

We prove that for any finitely generated group $G$ and any $k\geq1$, the space of $k$-colorings of $G$ does not admit a strict self-embedding. This settles the Gottschalk surjunctivity conjecture and, consequently, Kaplansky's direct finiteness conjecture.