Seminar on Combinatorics, Lie Theory, and Topology
# Stable finiteness of group algebras of surjunctive groups and model theory

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Europe/Rome

P1Ceccherini-SilbersteinCeccherini-Silberstein (Polo Fibonacci)
### P1Ceccherini-SilbersteinCeccherini-Silberstein

#### Polo Fibonacci

Description

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