Kazhdan constants of group extensions

TL;DR

This paper provides bounds and calculations for Kazhdan constants in various group extensions, improving existing results for certain classes of groups and automorphism groups.

Contribution

It introduces new bounds on Kazhdan constants for abelian group extensions and computes these constants for specific automorphism groups of nilpotent groups.

Findings

Improved bounds for Kazhdan constants of some meta-abelian groups

Calculated Kazhdan constants for tame automorphism groups of free nilpotent groups

Extended understanding of Kazhdan constants in group extension contexts

Abstract

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of $p$-groups of lower $p$-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.