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The algebraic cheap rebuilding property

Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold

TL;DR

The paper develops an axiomatic, algebraic framework to derive combination theorems for homological properties of groups and chain complexes via bootstrapping. It introduces the algebraic cheap rebuilding property (and a weaker version) and proves they are bootstrappable, unifying results on algebraic finiteness, $\ell^2$-invariants, Novikov--Shubin invariants, and vanishing torsion growth. By translating the geometric rebuilding approach of ABért–Bergeron–Frączyk–Gaboriau into an algebraic setting, it yields new vanishing results for torsion growth in graphs of groups with amenable vertex groups and elementary amenable edge groups, and it extends to amenable and arithmetic groups through quantified rebuildings. The framework provides robust, quantitative tools (mapping cones, projective replacements, and rebuildings) to control torsion and spectral invariants across residual chains, with broad applicability to $\mathsf{FP}_n$ and related finiteness properties. Overall, the work offers a cohesive, algebraic pathway to derive deep vanishing results for homological invariants in expansive group-theoretic constructions.

Abstract

We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $\ell^2$-invisibility, $\ell^2$-acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. As a key example, we introduce an algebraic version of Abért--Bergeron--Frączyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and fits into our axiomatic framework for combination theorems. In particular, we obtain that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.

The algebraic cheap rebuilding property

TL;DR

The paper develops an axiomatic, algebraic framework to derive combination theorems for homological properties of groups and chain complexes via bootstrapping. It introduces the algebraic cheap rebuilding property (and a weaker version) and proves they are bootstrappable, unifying results on algebraic finiteness, -invariants, Novikov--Shubin invariants, and vanishing torsion growth. By translating the geometric rebuilding approach of ABért–Bergeron–Frączyk–Gaboriau into an algebraic setting, it yields new vanishing results for torsion growth in graphs of groups with amenable vertex groups and elementary amenable edge groups, and it extends to amenable and arithmetic groups through quantified rebuildings. The framework provides robust, quantitative tools (mapping cones, projective replacements, and rebuildings) to control torsion and spectral invariants across residual chains, with broad applicability to and related finiteness properties. Overall, the work offers a cohesive, algebraic pathway to derive deep vanishing results for homological invariants in expansive group-theoretic constructions.

Abstract

We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, -invisibility, -acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. As a key example, we introduce an algebraic version of Abért--Bergeron--Frączyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and fits into our axiomatic framework for combination theorems. In particular, we obtain that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.
Paper Structure (19 sections, 36 theorems, 134 equations)

This paper contains 19 sections, 36 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on chain complexes
  3. Mapping cones
  4. Projective replacements
  5. Bootstrapping
  6. Bootstrappable properties of chain complexes
  7. Bootstrappable properties of groups and the Bootstrapping Theorem
  8. Examples of bootstrappable properties
  9. Algebraic finiteness properties
  10. l²-Invariants
  11. $\ell^2$-Invisibility
  12. $\ell^2$-Acyclicity
  13. Novikov--Shubin invariants and capacity
  14. $\ell^2$-Torsion
  15. Vanishing of (torsion) homology growth
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathsf{B}_*$ be a bootstrappable property of groups. Let $\Gamma$ be a group and let $n\in \mathbb{N}$. Let $\Omega$ be a $\Gamma$-CW-complex such that the following hold: Then $\Gamma\in \mathsf{B}_n$.

Theorems & Definitions (98)

  • Theorem 1.1: Bootstrapping Theorem \ref{['thm:bootstrapping']}
  • Definition 1.2
  • Theorem 1.3: Abért--Bergeron--Frączyk--Gaboriau
  • Theorem 1.4: Theorem \ref{['thm:bootstrapping CWR']}
  • Corollary 1.5: Corollary \ref{['cor:amenable graph']}
  • Definition 2.1: Mapping cone
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 88 more