From uncountable abelian groups to uncountable nonabelian groups

Uncountable Groups with Property (fh)

We exhibit some uncountable groups with Property (FH). In particular, these groups do not have Kazhdan’s Property (T), which is known to be equivalent to Property (FH) for countable groups. Our first examples rely on a theorem of Delzant, which states that every countable group embeds in a group with Property (T). We give two constructions. The first is a (non-explicit) transfinite induction on...

Uncountable groups have many nonconjugate subgroups

We prove that any uncountable group G of power ~, has at least 2 subgroups not conjugate in pairs. The paper is very self-contained, assuming no knowledge except cardinal arithmetic (and the definition of an (abelian) group).

An uncountable family of three-generated groups

3 Universal group of the family 5 3.1 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Overgroup G̃ and conjugacy classes of Gw . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Isomorphism classes . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Inert subgroups of uncountable locally finite groups

Let G be an uncountable universal locally finite group. We study subgroups H < G such that for every g ∈ G, |H : H ∩H| < |H|.

ON FREE PROFINITE GROUPS OF UNCOUNTABLE RANK∗ by