Menachem Shlossberg

Minimality of the inner automorphism group.

(Based on joint work with Dekui Peng)

By [1], a minimal group G is called z-minimal if G/Z(G) is minimal. In this paper, we present the z-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group G, let \mathrm{Inn}(G) be the group of all inner automorphisms of G, endowed with the Birkhoff topology. Using a theorem by Goto [2], we obtain our main result which asserts that if G is a connected Lie group and H\in\{G/Z(G), \mathrm{Inn}(G)\}, then H is minimal if and only if it is centre-free and topologically isomorphic to \mathrm{Inn}(G/Z(G)). In particular, if G is a connected Lie group with discrete centre, then \mathrm{Inn}(G) is minimal. We prove that a connected locally compact nilpotent group is z-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian z-minimal Lie group that is neither compact nor abelian.

[1] D. Dikranjan, W. He, D. Peng, W. Xi, Z. Xiao, Products of locally minimal groups, Topol. Appl. 329 (2023), https://doi.org/10.1016/j.topol.2022.108368

[2] M. Goto, Absolutely closed Lie groups, Math. Ann. 204 (1973), 337—341.