Between primitive and $2$-transitive: Synchronization and its friends

TL;DR

This paper explores the hierarchy of automorphism groups related to synchronization in automata, connecting group theory, combinatorics, and geometry, and discusses implications for the cernfd conjecture and open problems.

Contribution

It provides a unified overview of classes of groups between primitive and 2-homogeneous, introduces new results on the cernfd conjecture, and discusses interdisciplinary connections and open problems.

Findings

Hierarchy of synchronizing groups between primitive and 2-homogeneous.

New result related to the cernfd conjecture.

Identification of open problems and interdisciplinary links.

Abstract

An automaton is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the \v{C}ern\'y conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing $n$-state automaton has a reset word of length at most $(n-1)^2$. The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group $G$ on a set $\Omega$ is said to synchronize a map $f$ if the monoid $\langle G,f\rangle$ generated by $G$ and $f$ is synchronizing in the above sense; we say $G$ is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and $2$-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the \v{C}ern\'y conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present results that show the connections between the various areas of mathematics mentioned above, we include a new result on the \v{C}ern\'y conjecture, some challenges to finite geometers, some thoughts about infinite analogues, and a long list of open problems.