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$L^2$-torsion of automorphisms

Sam Hughes, Wolfgang Lueck

TL;DR

This work develops a comprehensive framework for $L^2$-torsion of group automorphisms, including a general blow-up/assembly formula that expresses $ ho^{(2)}(G)$ in terms of stabilisers for actions on contractible spaces, under determinant conjecture hypotheses. It then defines and analyzes the $L^2$-torsion of a self-homotopy equivalence and of a group automorphism, establishing essential invariance and multiplicativity properties, and applying them to a broad class of groups. The authors compute or vanishing-results for automorphisms in many geometrically significant settings—one-ended hyperbolic groups, relatively hyperbolic groups, RAAGs/RACGs, higher-dimensional graph manifolds, handlebody groups, and polynomially growing automorphisms—often reducing to the analysis of JSJ data or vertex stabilisers. They obtain concrete vanishing criteria for $ ho^{(2)}$ across CAT(0) lattices, graph manifolds, and other classes, highlighting the deep connections between $L^2$-torsion, group actions on spaces, and group decomposition theory. The results extend the understanding of $L^2$-torsion as a volume-like invariant, with implications for group stability, growth phenomena, and geometric group theory computations.

Abstract

We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a combination formula for the $L^2$-torsion of a group in terms of the $L^2$-torsion of its stabilisers of a sufficiently nice action on a contractible space. We apply it to compute the $L^2$-torsion of a selection of CAT(0) lattices, of many relatively hyperbolic groups and their automorphisms, of higher dimensional graph manifolds, and of handlebody groups.

$L^2$-torsion of automorphisms

TL;DR

This work develops a comprehensive framework for -torsion of group automorphisms, including a general blow-up/assembly formula that expresses in terms of stabilisers for actions on contractible spaces, under determinant conjecture hypotheses. It then defines and analyzes the -torsion of a self-homotopy equivalence and of a group automorphism, establishing essential invariance and multiplicativity properties, and applying them to a broad class of groups. The authors compute or vanishing-results for automorphisms in many geometrically significant settings—one-ended hyperbolic groups, relatively hyperbolic groups, RAAGs/RACGs, higher-dimensional graph manifolds, handlebody groups, and polynomially growing automorphisms—often reducing to the analysis of JSJ data or vertex stabilisers. They obtain concrete vanishing criteria for across CAT(0) lattices, graph manifolds, and other classes, highlighting the deep connections between -torsion, group actions on spaces, and group decomposition theory. The results extend the understanding of -torsion as a volume-like invariant, with implications for group stability, growth phenomena, and geometric group theory computations.

Abstract

We develop the theory of -torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a combination formula for the -torsion of a group in terms of the -torsion of its stabilisers of a sufficiently nice action on a contractible space. We apply it to compute the -torsion of a selection of CAT(0) lattices, of many relatively hyperbolic groups and their automorphisms, of higher dimensional graph manifolds, and of handlebody groups.
Paper Structure (18 sections, 32 theorems, 79 equations)

This paper contains 18 sections, 32 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Basics about L2-torsion
  4. Blowing up orbits by classifying spaces of families
  5. L2-torsion of a selfhomotopy equivalence
  6. L2-torsion of a group automorphism
  7. A short discussion of the finiteness assumptions
  8. Computations
  9. Lattices
  10. Higher dimensional graph manifolds
  11. Relatively hyperbolic groups
  12. Automorphisms of one-ended relatively hyperbolic groups
  13. Polynomially growing automorphisms
  14. Free products
  15. Relatively hyperbolic groups
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $G$ be a group acting cocompactly on a contractible $CW$-complex $X$ such that the fixed point sets of finite subgroups of $G$ are contractible. Suppose that each cell stabiliser $H_\sigma$ of the action of $G$ is $L^2$-acyclic and admits a finite model for $\underline{E}H_\sigma$. If $G$ satisf

Theorems & Definitions (83)

  • Conjecture 1.1: Homological torsion growth and $L^2$-torsion
  • Conjecture 1.2: Modified Homological torsion growth and $L^2$-torsion
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.2
  • Conjecture 1.3: Vanishing of the $L^2$-torsion of an automorphism of subexponential growth
  • Remark 1.4
  • Lemma 2.1
  • proof
  • ...and 73 more