Hilbert space compression under direct limits and certain group extensions

TL;DR

This paper investigates how Hilbert space compression behaves under direct limits of groups and provides bounds for group extensions involving hyperbolic groups or groups with polynomial growth.

Contribution

It introduces new bounds on Hilbert space compression for limits of groups and estimates for extensions with hyperbolic or polynomial growth groups.

Findings

Bounds on Hilbert space compression for directed limits of groups

Estimates for Hilbert space compression of certain group extensions

Insights into the behavior of group properties under limits and extensions

Abstract

We find bounds on the Hilbert space compression of the limit of a directed metric system of groups. We also give estimates on the Hilbert space compression of group extensions of a group $H$ by a a word-hyperbolic group or a group of polynomial growth.