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The product formula for Reidemeister numbers on nilpotent groups

Pieter Senden

TL;DR

The paper analyzes Reidemeister numbers $R(\varphi)$ in finitely generated torsion-free nilpotent groups through central-extension methods and a generalized product formula. It derives an addition formula for central extensions and shows that, for residually finite groups of type $(F)$ with a central subgroup $C$, one can have $R(\varphi)=R(\varphi|_{C})R(\bar{\varphi})$ under certain conditions. The authors use these results to establish criteria for when $R(\varphi)$ is infinite, relate twisted stabilisers to fixed points, and demonstrate that Reidemeister numbers behave predictably under finite-index subgroups in FGTF nilpotent groups. As a key outcome, they construct infinite families of groups with full Reidemeister spectrum by transferring spectrum from subgroups to finite-index extensions, including new instances built from free nilpotent structures such as $N_{r,2}(m)$.

Abstract

We study the product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups in two ways. On the one hand, we generalise the product formula to central extensions. On the other hand, we derive general results for finitely generated (torsion-free) nilpotent groups from the product formula: we provide a strong tool to prove that an endomorphism on a finitely generated nilpotent group has infinite Reidemeister number and compute Reidemeister numbers on finite index subgroups of finitely generated torsion-free nilpotent groups. Using the latter, we provide an infinite family of groups with full Reidemeister spectrum.

The product formula for Reidemeister numbers on nilpotent groups

TL;DR

The paper analyzes Reidemeister numbers in finitely generated torsion-free nilpotent groups through central-extension methods and a generalized product formula. It derives an addition formula for central extensions and shows that, for residually finite groups of type with a central subgroup , one can have under certain conditions. The authors use these results to establish criteria for when is infinite, relate twisted stabilisers to fixed points, and demonstrate that Reidemeister numbers behave predictably under finite-index subgroups in FGTF nilpotent groups. As a key outcome, they construct infinite families of groups with full Reidemeister spectrum by transferring spectrum from subgroups to finite-index extensions, including new instances built from free nilpotent structures such as .

Abstract

We study the product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups in two ways. On the one hand, we generalise the product formula to central extensions. On the other hand, we derive general results for finitely generated (torsion-free) nilpotent groups from the product formula: we provide a strong tool to prove that an endomorphism on a finitely generated nilpotent group has infinite Reidemeister number and compute Reidemeister numbers on finite index subgroups of finitely generated torsion-free nilpotent groups. Using the latter, we provide an infinite family of groups with full Reidemeister spectrum.

Paper Structure

This paper contains 8 sections, 35 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Central extensions
  3. Addition formula for central extensions
  4. Product formula for central extensions of residually finite groups
  5. Three applications of the product formula
  6. Application 1: a sufficient condition for infinite Reidemeister number
  7. Application 2: twisted stabilisers in nilpotent groups
  8. Application 3: Reidemeister numbers on finite index subgroups

Key Result

Theorem 1.1

Let $N$ be a finitely generated torsion-free nilpotent group and let $\varphi \in \mathop{\mathrm{End}}\nolimits(N)$. Suppose that is a central series of $N$ such that Let $\varphi_{i}$ denote the induced endomorphism on $\frac{N_{i}}{N_{i - 1}}$ for $i \in \{1, \ldots, c\}$. Then In particular, $R(\varphi) = \infty$ if and only if $R(\varphi_{i}) = \infty$ for some $i \in \{1, \ldots, c\}$

Theorems & Definitions (64)

  • Theorem 1.1: Product formula, Romankov11
  • Proposition 1.2: Romankov11
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3: KimLeeLee05
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • ...and 54 more