Permanence properties of the second nilpotent product of groups

ROMAN SASYK

BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN

BELGIAN MATHEMATICAL SOC TRIOMPHE

Año: 2019 vol. 26 p. 725 - 742

1370-1444

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.