Stable finiteness of group algebras of surjunctive groups and model theory

by Tullio Ceccherini-Silberstein (Università del Sannio)

Thursday 20 Jun 2024, 14:45 → 15:45 Europe/Rome

P1Ceccherini-SilbersteinCeccherini-Silberstein (Polo Fibonacci)

Let G be a group and let A be a finite set. Equip A^G = \{x \colon G \to A\} with the
prodiscrete topology and with the G-action  given by (gx)(h) = x(g^{-1}h) for all x \in A^G and g,h \in G.
  A continuous G-equivariant map \tau \colon A^G \to A^G is called a cellular automaton.
  One says that G is surjunctive provided that every injective cellular automaton is surjective
(and therefore a homeomorphism). Gotthshalk conjectured that all groups are surjunctive. Gromov and, independently,
B. Weiss proved that all sofic groups (and therefore all residually finite and all amenable groups) are surjunctive.
  A ring R is said to be stably finite provided that for every integer d \geq 1 the algebra Mat_d(R)
of d \times d matrices with coefficients in R satisfies the following condition: if AB = I then BA = I, for all A,B \in Mat_d(R).
Kaplansky showed that group algebras with coefficients in a field of characteristic 0 are stably finite and conjectured
that the same holds also in positive characteristic.
  Using algebraic geometry methods, Xuan Kien Phung has shown that the group algebra of a surjunctive group
is stably finite. In other words, every group satisfying Gottschalk's surjunctivity conjecture also satisfies
Kaplansky's  stable finiteness conjecture.
  I will explain a proof of this result based on first-order model theory.
This is work in collaboration with Michel Coornaert and Phung