# Permanence properties of the second nilpotent product of groups

## TL;DR

This paper investigates how key properties like amenability, property (T), and exactness are preserved under the second nilpotent product of groups, introduces a related wreath product, and explores its properties.

## Contribution

It establishes the preservation of several important group properties under the second nilpotent product and defines a new restricted wreath product with its properties.

## Key findings

- Amenability, property (T), and exactness are preserved under second nilpotent product.
- The restricted second nilpotent wreath product retains the Haagerup property if the original groups have it.
- Unitarizability of the wreath product from abelian groups depends on the amenability of the acting group.

## Abstract

We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath product constructed from it is unitarizable if and only if the acting group is amenable.