Minimality of topological matrix groups and Fermat primes

M. Megrelishvili, M. Shlossberg

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group ST+(n,F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL(n,F), where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p−1,Q) is minimal, where Q is the field of rationals equipped with the p-adic topology; (3) SL(p−1,Q(i)) is minimal, where Q(i)⊂C is the Gaussian rational field.

Original languageEnglish
Article number108272
JournalTopology and its Applications
Volume322
DOIs
StatePublished - 1 Dec 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier B.V.

Funding

This research was supported by a grant of the Israel Science Foundation ( ISF 1194/19 ) and also by the Gelbart Research Institute at the Department of Mathematics, Bar-Ilan University .

FundersFunder number
Gelbart Research Institute at the Department of Mathematics, Bar-Ilan University
Israel Science FoundationISF 1194/19

    Keywords

    • Fermat primes
    • Iwasawa decomposition
    • Local field
    • Matrix group
    • Minimal topological group
    • Projective linear group
    • Special linear group

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