Around Gromov's injectivity lemma and applications to post-injunctive groups

TL;DR

This paper explores the extension of Gromov's injectivity lemma to properties like post-surjectivity and pre-injectivity in cellular automata, and studies the stability of post-injunctive groups, especially in relation to sofic and residually finite groups.

Contribution

It generalizes Gromov's lemma to other dynamical properties and investigates the stability of post-injunctive groups under group extensions.

Findings

Post-surjective cellular automata over sofic groups are bijective.

Semidirect extensions of post-injunctive groups with residually finite kernels are also post-injunctive.

Analogous results to Gromov's lemma are established for properties beyond injectivity.

Abstract

Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective cellular automata are injective. However, if the surjectivity condition is replaced by a certain strengthened property called post-surjectivity then all post-surjective cellular automata must be bijective whenever the universe is a sofic group. A group universe is said to be post-injunctive if every post-surjective cellular automaton with finite alphabet over this group universe must be bijective. Gromov's injectivity lemma states each injective cellular automaton over a subshift can be extended to an injective cellular automaton over every subshift which is close enough to the initial subshift. In this paper, we obtain analogous results where injectivity is replaced by other fundamental dynamical properties namely post-surjectivity and pre-injectivity. We also study various stable properties of the class of post-injunctive groups in parallel to properties of surjunctive groups. Among the results, we show that semidirect extensions of post-injunctive groups with residually finite kernels must be post-injunctive.