Let G be a group and let V be a finite-dimensional vector space over a field K. We equip VG = {x : G → V} with the prodiscrete uniform structure, the G-shift action ((gx)(h) := x(g-1h)), and the natural structure of a K-vector space. A G-invariant closed subspace X ≤ VG is called a linear subshift. A linear subshift X ⊆ VG is said to be of finite type provided that there exists a finite subset Ω ⊆ G and a subspace W ⊆ VΩ such that
X = X(Ω,W) := {x ∈ VG : (gx)|Ω ∈ W for all g ∈ G}.
The group G is said to be of linear-Markov type if for every finite-dimensional vector space V over any field K, every linear subshift X ⊆ VG is of finite type. A uniformly continuous and G-equivariant K-linear map τ : VG → VG is called a cellular automaton. The group G is said to be linearly surjunctive provided that for every finite-dimensional vector space V over any field K the following holds: every injective linear cellular automaton τ : VG → VG is surjective.
THEOREM 1 (CS-Coornaert 2007) Sofic groups are linearly surjunctive.
COROLLARY 1 (Elek-Szabo 2004; CS-Coornaert 2007) Group rings of sofic groups are stably finite.
THEOREM 2 (CS-Coornaert-Phung 2020) A group is of linear-Markov type if and only if the group ring K[G] is left-Noetherian for any field K.
COROLLARY 2 (CS-Coornaert-Phung 2020) Polycyclic-by-finite groups are of linearly-Markov type.
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
