Somewhere trivial automorphisms

共轭 $\mathcal P(\mathbb N)/\mathrm{Fin}$ 的平凡自同构

Conjugating trivial automorphisms of $\mathcal P(\mathbb N)/\mathrm{Fin}$

本文研究 $\mathcal P(\mathbb N)/\mathrm{Fin}$ 的平凡自同构的共轭性，在连续统假设下给出无一阶和指标障碍的充要条件，并应用于证明粗空间的一致 Roe 冠状同构独立于 ZFC。

Trivial automorphisms

We prove that the statement `For all Borel ideals I and J on $ω$, every isomorphism between Boolean algebras $P(ω)/I$ and $P(ω)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(ω)/I$ and any other quotient $P(ω)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $ω$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(ω)/Fin$ are trivial. Proofs rely on delicate analysis of name…

Automorphisms of $\mathcal P(ω)/\mbox{fin}$ and large continuum

We prove that it is consistent with $\mathfrak c>\aleph_2$ that all automorphisms of $\mathcal P(ω)/\mbox{fin}$ are trivial.

约化积之间的平凡同构

Trivial Isomorphisms between Reduced Products

本文提出在弱力迫公理下证明可数模型约化积自同构极小化的通用方法，并推出域、序、树、随机图及 P(N)/Fin 的自同构均平凡。

Comparing the automorphism group of the measure algebra with some groups related to the infinite permutation group of the natural numbers

We prove, by a straight construction, that the automorphism group of the measure algebra and the subgroup of the measure preserving ones cannot be isomorphic to the trivial automorphisms of P(N)/fin.

论 $\mathcal P(λ)/[λ]^{<λ}$ 的自同构

On automorphisms of $\mathcal P(λ)/[λ]^{<λ}$

本文研究商布尔代数 $\mathcal P(λ)/[λ]^{<λ}$ 的自同构平凡性，分别在 MA、可测基数及不可达基数的力迫情形下给出一致性与反例结果。

Automorphism groups of linearly ordered homogeneous structures

We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $n\geq 2$, and prove the simplicity of their automorphism groups.

Universal flows and automorphisms of $\mathcal P(ω)/\mathrm{fin}$

We prove that for every countable discrete group $G$, there is a $G$-flow on $ω^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of weight $\leq\!\mathfrak{c}$. Applying Stone duality, we deduce that, under \mathsf{CH}, there is a trivial automorphism $τ$ of $\mathcal P(ω)/\mathrm{fin}$ with every other automorphism embedded in it, which means that every other automorphism of $\mathcal P(ω)/\mathrm{fin}$ can be written as the restriction of $τ$ to a suitably chosen subalgebra.

Indecomposable explicit abelian group

(withdrawn.) For every lambda we give an explicit construction of an Abelian group with no non-trivial automorphisms. In particular the group absolutely has no non-trivial automorphisms, hence is absolutely indecomposable. Earlier we knew a stronger existence theorem but only up to a quite large cardinal which was a necessary restriction. In another direction the construction does not use the axiom of choice.

The height of the automorphism tower of a group

For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We first show that for any kappa=kappa^{aleph_0} there is a group of cardinality kappa with height>kappa^+ (improving the lower bound). Second we show that for many such kappa there is a group of height > 2^kappa, so proving that the upper bound essentially cannot be improved.