Amenability of Bounded Automata Groups on Infinite Alphabets

TL;DR

This paper establishes an amenability criterion for groups generated by bounded automata with infinite alphabets, extending known results from finite alphabet cases and motivated by iterated monodromy groups of entire functions.

Contribution

It introduces a new amenability criterion for bounded automata groups on infinite alphabets based on recurrence, generalizing finite alphabet results.

Findings

The criterion links amenability to recurrence of the first level action.

All bounded activity automata groups with finite alphabets are amenable.

The work is motivated by iterated monodromy groups of entire functions.

Abstract

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.