# Universality of group embeddability

## TL;DR

This paper demonstrates that various types of embeddability among groups are invariantly universal analytic quasi-orders, unifying and strengthening previous results in the field of Borel reducibility.

## Contribution

It establishes the invariance of universality for embeddability notions across countable, Polish, and separable groups, advancing the understanding of their complexity.

## Key findings

- Embeddability between countable groups is invariantly universal.
- Topological embeddability among Polish groups is invariantly universal.
- Isometric embeddability among separable groups with bounded bi-invariant metrics is invariantly universal.

## Abstract

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the isometric embeddability between separable groups with a bounded bi-invariant complete metric are all invariantly universal analytic quasi-orders. This strengthens some results from [Wil14] and [FLR09].