ScholarLens Back to search
2022•arXiv (Cornell University)Open access

On upper bounds for the first $\ell^2$-Betti number

Carsten Feldkamp, Steffen Kionke

Open full text 0 citations

Abstract

This article presents a method for proving upper bounds for the first $\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first $\ell^2$-Betti number. Our approach extends to generalizations of $\ell^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Open-access reader

About this research paper

What this paper is about

This article presents a method for proving upper bounds for the first $\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first $\ell^2$-Betti number. Our approach extends to generalizations of $\ell^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Why it matters

A significance statement is not available in the OpenAlex record.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

This article presents a method for proving upper bounds for the first $\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first $\ell^2$-Betti number. Our approach extends to generalizations of $\ell^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Key concepts: Betti number, Mathematics, Combinatorics, Exponent, Upper and lower bounds, Graph, Prime (order theory), Discrete mathematics

Related papers

Related paper 1Related paper 2
Back to paper searchBrowse research topicsOriginal source
On upper bounds for the first $\ell^2$-Betti number — Research Paper | ScholarLens