Michael Megrelishvili

Topological minimality of matrix groups and Fermat primes

In the first part I will present several old and new results about minimality properties of topological groups. Some of them are inspired by [2].
In the second part I will discuss topological minimality of some natural matrix groups defined over subfields of local fields. The main results are based on a joint published work (2022) with Menachem Shlossberg [3] which is dedicated to Prof. D. Dikranjan on the occasion of his 70th birthday.
We show that the special upper triangular group \mathrm{ST^+}(n,\mathbb F)
is minimal for every local field \mathbb F of characteristic \neq 2. This result is new even for the field \mathbb R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group \mathrm{SL}(n,\mathbb F), where \mathbb F is a subfield of a local field. This extends some known results of Remus-Stoyanov [4] and Bader–Gelander [1].
One of our main applications is a characterization of Fermat primes in terms of minimality, which asserts that for an odd prime p the following conditions are equivalent:
– p is a Fermat prime;
– \mathrm{SL}(p-1,\mathbb Q) is minimal, where \mathbb Q is the field of rationals equipped with the p-adic topology;
– \mathrm{SL}(p-1,\mathbb Q(i)) is minimal, where \mathbb Q(i) \subset \mathbb C is the Gaussian rational field.

References
[1] U. Bader, T. Gelander, Equicontinuous actions of semisimple groups, Groups, Geometry and Dynamics 11 (2017).
[2] D. Dikranjan and M. Megrelishvili, Minimality Conditions in Topological Groups, in: Recent Progress in General Topology III, 229–327, K.P. Hart, J. van Mill, P. Simon (Eds.), Springer, Atlantis Press, 2014.
[3] M. Megrelishvili and M. Shlossberg, Minimality of topological matrix groups and Fermat primes, Topology Appl. 322 (2023).
[4] D. Remus, L. Stoyanov, Complete minimal topological groups, Topology Appl. 42 (1991).