Equi-variant and Stable Finite Decomposition Complexity

TL;DR

This paper explores different types of decomposition complexity in groups, characterizes residually finite groups with finite decomposition complexity, and introduces equi-variant and stable FDC, linking group properties to their box spaces.

Contribution

It introduces equi-variant straight FDC and stable FDC, providing new characterizations and extending understanding of decomposition complexity in group theory.

Findings

Residually finite groups with FDC characterized

Equi-variant sFDC equivalent to box space sFDC

Elementary amenable groups have equi-variant sFDC

Abstract

In this paper, we introduce and study various kinds of decomposition complexity. First, we give a characterization of residually finite groups having finite decomposition complexity (FDC). Secondly, we introduce equi-variant straight FDC (sFDC), and prove that a group having equi-variant sFDC if and only if its box space having sFDC. Finally, we show that elementary amenable groups have equi-variant sFDC by introducing something called stable FDC.