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On a coarse invertibility spectrum for coarse groups

Leo Schäfer, Federico Vigolo

TL;DR

The paper introduces the coarse power-invertibility spectrum, a coarse invariant given by the set of exponents $n$ for which the $n$-th power map is a coarse equivalence, and proves its invariance under coarse isomorphism. It computes this spectrum for two families of coarse groups on $\mathbb{Z}$: word-metric coarsifications $(\mathbb{Z},\mathcal{E}_{\mathrm{Cay}(S_g)})$ and profinite coarsifications $(\mathbb{Z},\mathcal{E}_Q)$. For the Cayley-based metrics, the spectrum equals the set of primes dividing $g$, implying $\mathbb{Z}$ with $d_{g_1}$ and $d_{g_2}$ are not coarsely isomorphic unless $g_1$ and $g_2$ share the same prime divisors; in particular $2$ and $3$ are not coarsely isomorphic. For profinite coarsifications, the spectrum recovers the defining prime set $Q$, yielding a complete coarse-distinction among these coarsifications. The results demonstrate the effectiveness of a coarse-algebraic invariant in separating coarse groups where purely geometric methods are inconclusive.

Abstract

We introduce a coarse algebraic invariant for coarse groups and use it to differentiate various coarsifications of the group of integers. This lets us answer two questions posed by Leitner and the second author. The invariant is obtained by considering the set of exponents n such that taking n-th powers defines a coarse equivalence of the coarse group.

On a coarse invertibility spectrum for coarse groups

TL;DR

The paper introduces the coarse power-invertibility spectrum, a coarse invariant given by the set of exponents for which the -th power map is a coarse equivalence, and proves its invariance under coarse isomorphism. It computes this spectrum for two families of coarse groups on : word-metric coarsifications and profinite coarsifications . For the Cayley-based metrics, the spectrum equals the set of primes dividing , implying with and are not coarsely isomorphic unless and share the same prime divisors; in particular and are not coarsely isomorphic. For profinite coarsifications, the spectrum recovers the defining prime set , yielding a complete coarse-distinction among these coarsifications. The results demonstrate the effectiveness of a coarse-algebraic invariant in separating coarse groups where purely geometric methods are inconclusive.

Abstract

We introduce a coarse algebraic invariant for coarse groups and use it to differentiate various coarsifications of the group of integers. This lets us answer two questions posed by Leitner and the second author. The invariant is obtained by considering the set of exponents n such that taking n-th powers defines a coarse equivalence of the coarse group.
Paper Structure (9 sections, 16 theorems, 31 equations)

This paper contains 9 sections, 16 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. A number-theoretic problem
  3. Context: coarse groups
  4. Technique of proof
  5. Spectra of power invertibility
  6. Power invertibility spectra for certain word-metrics on $\mathbb Z$
  7. Power invertibility spectra of profinite coarsifications of $\mathbb Z$
  8. Open Problems
  9. Appendix: coarse vs. bi-Lipschitz equivalence

Key Result

Theorem 1.1

Given $g_1,g_2\geq 2$ the identity map $(\mathbb Z,d_{g_1})\to (\mathbb Z,d_{g_2})$ is a bi-Lipschitz equivalence if and only if $g_1^n=g_2^m$ for some $n,m\in\mathbb N$.

Theorems & Definitions (34)

  • Theorem 1.1: nathanson2011bi
  • Theorem 1
  • Theorem 2
  • Remark 1.3
  • Remark 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 24 more