Rational Reidemeister trace of an outer automorphism of finite order

Smrekar, Jaka (author)

Let $H$ be a group of type $FP$ over the rationals and let $\phi\colon H\to H$ be an automorphism whose class in ${\rm Out}(H)$ has finite order and such that the Lefschetz number of some iterate $\phi^i$ is nonzero. We show that under a suitable Bass conjecture assumption, this iterate $\phi^i$ is conjugate to a finite order automorphism, and the intersection of the fixed point set of $\phi$ with the center of $H$ is of bounded exponent. In the torsion-free case it follows that the center of the HNN extension $\langle H,t\,\vert\,\forall h\in H:\,tht^{-1}=\phi(h)\rangle$ is infinite cyclic. These results are related to Geoghegan and Nicas' first order Euler characteristic of a group $G$ with a finite Eilenberg-MacLane complex; this is a morphism $\chi_1(G)$ from the center of $G$ to $H_1(G;k)$ where $k$ is a ring of coefficients. We give a complete determination of $\chi_1(G)$ with rational coefficients.

• Language: English
• Keywords: automorphism of finite order, Reidemeister trace, Hattori-Stallings rank, Gottlieb’s theorem, first order Euler characteristic, Cooke’s obstruction
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: 13 str.
• Numbering: Vol. 227, iss. 7, art. 107350
• PID: 20.500.12556/RUL-148644
• UDC: 512.54
• ISSN on article: 0022-4049
• DOI: 10.1016/j.jpaa.2023.107350
• COBISS.SI-ID: 141555715
• Publication date in RUL: 28.08.2023

SMREKAR, Jaka, 2023, Rational Reidemeister trace of an outer automorphism of finite order. Journal of pure and applied algebra [online]. 2023. Vol. 227, no. 7,  107350. [Accessed 17 May 2025]. DOI 10.1016/j.jpaa.2023.107350. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=148644

Record is a part of a journal

• Title: Journal of pure and applied algebra
• Shortened title: J. pure appl. algebra
• Publisher: Elsevier
• ISSN: 0022-4049
• COBISS.SI-ID: 4746506

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: avtomorfizem končnega reda, Reidemeistrova sled, Hattori-Stallingsov rang, Gottliebov izrek, Eulerjeva karakteristika prvega reda, Cookova ovira

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: N1-0237
• Name: Holomorfne parcialne diferencialne relacije

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0291
• Name: Analiza in geometrija