Right Amenability And Growth Of Finitely Right Generated Left Group Sets

TL;DR

This paper introduces new concepts for analyzing left homogeneous spaces with coordinate systems, characterizes right amenability via isoperimetric constants, and links sub-exponential growth to right amenability, including quotient sets of groups.

Contribution

It defines right generating sets and related graph invariants, and establishes a characterization of right amenability in terms of isoperimetric constants and growth rates.

Findings

Right amenability corresponds to zero isoperimetric constant.

Sub-exponential growth implies right amenability for spaces with finite stabilizers.

Quotients of groups with sub-exponential growth by finite subgroups are right amenable.

Abstract

We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable.