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Detecting virtual homomorphisms via Banach metrics

Liran Ron-George, Ariel Yadin

TL;DR

The paper addresses the problem of detecting virtual homomorphisms from a finitely generated infinite group $G$ to $\mathbb{Z}$ by examining metric-functionals at infinity. It introduces Banach metrics, a flexible generalization of Cayley metrics, and proves that $G$ admits a virtual homomorphism if and only if there exists a Banach metric on $G$ whose metric-functional boundary $\partial(G,d)$ contains a finite orbit; when a genuine homomorphism to $\mathbb{Z}$ exists, the corresponding boundary function can be chosen as a fixed point. Banach metrics are shown to behave well under passage to finite-index subgroups, enabling transfer of boundary-capture results to subgroups. The paper also analyzes the limitations of Cayley metrics in detecting virtual homomorphisms, proves a no-detection result for non-virtually cyclic Gromov hyperbolic groups, and establishes positive results for virtually nilpotent (and virtually Abelian) groups. Overall, Banach metrics provide a robust framework connecting boundary dynamics, group growth, and the existence of virtual homomorphisms, with potential implications for Grigorchuk’s gap conjecture and related questions on growth and indicability.

Abstract

We introduce the notion of "Banach metrics" on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into Z, the additive group of integers. We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn't always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.

Detecting virtual homomorphisms via Banach metrics

TL;DR

The paper addresses the problem of detecting virtual homomorphisms from a finitely generated infinite group to by examining metric-functionals at infinity. It introduces Banach metrics, a flexible generalization of Cayley metrics, and proves that admits a virtual homomorphism if and only if there exists a Banach metric on whose metric-functional boundary contains a finite orbit; when a genuine homomorphism to exists, the corresponding boundary function can be chosen as a fixed point. Banach metrics are shown to behave well under passage to finite-index subgroups, enabling transfer of boundary-capture results to subgroups. The paper also analyzes the limitations of Cayley metrics in detecting virtual homomorphisms, proves a no-detection result for non-virtually cyclic Gromov hyperbolic groups, and establishes positive results for virtually nilpotent (and virtually Abelian) groups. Overall, Banach metrics provide a robust framework connecting boundary dynamics, group growth, and the existence of virtual homomorphisms, with potential implications for Grigorchuk’s gap conjecture and related questions on growth and indicability.

Abstract

We introduce the notion of "Banach metrics" on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into Z, the additive group of integers. We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn't always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.
Paper Structure (17 sections, 15 theorems, 40 equations, 1 figure)

This paper contains 17 sections, 15 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Metric-functionals
  4. Cayley graphs
  5. Action on the boundary
  6. Banach metrics
  7. Finite index subgroups
  8. Detection of virtual homomorphisms
  9. Nilpotent groups
  10. Gap conjecture
  11. Open questions
  12. Metric functionals and quotient groups
  13. Basic properties of metric functionals
  14. Banach metric construction
  15. Virtual homomorphisms
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $G$ be a finitely generated Gromov hyperbolic group, which is not virtually cyclic. Let $d_S$ be the metric of some Cayley graph on $G$. Then, there are no finite orbits in $\partial (G,d_S)$.

Figures (1)

  • Figure 1: The standard Cayley graph of $\mathbb{Z}^2$ with its metric-functional boundary.

Theorems & Definitions (36)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Conjecture 2.10: Grigorchuk's gap conjecture
  • ...and 26 more