On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb Z$ groups

Sam P. Fisher; Pablo Sánchez-Peralta

On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb Z$ groups

Group Theory 2026-01-27 v1

Authors: Sam P. Fisher , Pablo Sánchez-Peralta

Abstract

Let $G$ be a residually poly- $\mathbb Z$ group of finite type. We prove that $G$ admits a poly- $\mathbb Z$ quotient with kernel $N$ satisfying $\mathrm{cd}_{\mathbb Q}(N) < \mathbb{cd}_{\mathbb Q}(G)$ if and only if the top-dimensional $L^2$ -Betti number of $G$ vanishes.

Keywords

$L^2$-Betti number residually poly-$\mathbb Z$ groups cohomological dimension group quotient

Cite

@article{arxiv.2601.18594,
  title  = {On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb Z$ groups},
  author = {Sam P. Fisher and Pablo Sánchez-Peralta},
  journal= {arXiv preprint arXiv:2601.18594},
  year   = {2026}
}

Comments

9 pages. Intended for publication in the "Geometry and Topology of Polyhedral Complexes" conference proceedings

On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb Z$ groups

Abstract

Keywords

Cite

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