Extension properties of asymptotic property C and finite decomposition complexity

TL;DR

This paper establishes extension theorems for geometric properties like APC and FDC in finitely generated groups with quasi-actions, broadening understanding of their inheritance under weaker conditions.

Contribution

It introduces new extension theorems for properties such as APC and FDC under weak quasi-action assumptions, expanding the scope of geometric group theory.

Findings

Extension theorems for APC, FDC, sFDC in groups with quasi-actions.

Conditions under which properties are inherited by groups from quasi-stabilizers.

Demonstrates flexibility of weak quasi-action assumptions in geometric property extensions.

Abstract

We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group $G$ with a word metric and a coarse quasi-action on a metric space $X$. We assume that the quasi-stabilizers have a property $P_1$, and $X$ has the same or sometimes a weaker property $P_2$. Then $G$ also has property $P_2$. We show some sample applications, discuss constraints to further generalizations, and illustrate the flexibility that the weak quasi-action assumption allows.