Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2502.16651

The product formula for Reidemeister numbers on nilpotent groups

Pieter Senden
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Pieter Senden. 2025-02-23. The product formula for Reidemeister numbers on nilpotent groups. https://arxiv.org/abs/2502.16651

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗