Isometry types of profinite groups

Some model theory of profinite groups

We give some background on uniform pro-p groups and the model theory of profinite NIP groups.

On small profinite groups

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if every subgroup of finite index is open. We show that two profinite groups that are elementarily equivalent, in the first-order language of groups, are isomorphic if one of them is strongly complete, extending a result of Moshe Jarden and Alexander Lubotzky which treats the case of finitely generated profinite groups.

An axiomatizable profinite group with infinitely many open subgroups of index 2

We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.

Finite axiomatizability for profinite groups

A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed.

On strongly just infinite profinite branch groups

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As…

排序投射有限群的嵌入性质

The embedding property for sorted profinite groups

该研究在排序投射有限群范畴中引入排序嵌入性质SEP，证明其存在通用SEP覆盖，提出等价的有限排序嵌入性质FSEP并用于证明相关理论的ω-稳定性，还得到通用覆盖的唯一性，推广了经典投射有限群的相关结果。