Uniformly bounded fibred coarse embeddability and uniformly bounded a-T-menability

TL;DR

This paper introduces a new concept called uniformly bounded fibred coarse embeddability, explores its connection with uniformly bounded a-T-menability of groups, and provides examples highlighting differences from existing notions.

Contribution

The paper generalizes fibred coarse embeddability to a uniformly bounded setting and links it to a-T-menability, expanding understanding of metric space and group properties.

Findings

Introduces uniformly bounded fibred coarse embeddability.

Establishes its relationship with uniformly bounded a-T-menability.

Provides examples illustrating differences from fibred coarse embeddability.

Abstract

In this paper, we introduce the concept of uniformly bounded fibred coarse embeddability of metric spaces, generalizing the notion of fibred coarse embeddability defined by X. Chen, Q. Wang and G. Yu. Moreover, we show its relationship with uniformly bounded a-T-menability of groups. Finally, we give some examples to illustrate the differences between uniformly bounded fibred coarse embeddability and fibred coarse embeddability.