Order and minimality of some topological groups

Abstract

We study the minimality of some natural matrix groups defined on nice topological fields F. More precisely, we examine the minimality of the special upper triangular groups SUT(n, F), the special linear groups SL(n, F) and the projective general linear groups PGL(n, F). We prove that if F is a local field of characteristic different than 2, then SUT(n, F) is minimal for every n ∈ N. This result is new even for F = R and n = 2. In contrast, we show that SUT(3, Q(i)) is not minimal, where Q(i) is the Gaussian rational field.Using Iwasawa decomposition our technique leads to the total minimality of SL(n, F) for local fields F of characteristic different than 2. Note that recent results of Bader and Gelander [2], obtained in a different way, imply that the same is true for every characteristic. This extends the well known result of Remus and Stoyanov [19] about the total minimality of SL(n, R).Moreover, we show that if F is a local field, then PGL(n, F) is totally minimal for every n ∈ N.

Minimum topological group topologies

“…In this case the minimum group topology is the usual compact-open topology on H(X). Recently Megrelishvili and Polev [6] have extended this last theorem to H(X) for many compact linearly ordered spaces X.…”

“…We continue this line of investigation into the existence, or otherwise, of minimum Hausdorff group topologies. A number of questions raised in [2] are answered (notably, Questions 2.3 and 4.28), as are Questions 5.1, 5.2(1) and 5.3 (1) from [6].…”

C-Minimal topological groups

“…Example 1.7. (a) By Bader and Gelander [1], the special linear group SL(n, F) is totally minimal for every local field F (see also [23] for a new independent proof). In particular, SL…”

Minimality of the Semidirect Product

“…In [20] the two first-named authors study the minimality of the group H + (X), where X is a compact linearly ordered space and H + (X) is the topological group of all order-preserving homeomorphisms of X. In general, H + (X) need not be minimal.…”