Incomparable actions of free groups

Borel reductions of profinite actions of SL(n,Z)

Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank \emph{strictly increases} in complexity with the rank. Subsequently, Thomas proved that the complexity of the classification problems for $p$-local torsion-free abelian groups of fixed rank $n$ are \emph{pairwise incomparable} as $p$ varies. We prove that if $3\leq m<n$ and $p,q$ are distinct primes, then the complexity of the classification problem for $p$-local torsion-free abelian groups of rank $m$ is again incomparable with that for $q$-local torsion-free abelian groups of ra…

The classification of torsion-free abelian groups of finite rank up to isomorphism and up to quasi-isomorphism

The isomorphism and quasi-isomorphism relations on the $p$-local torsion-free abelian groups of rank $n\geq3$ are incomparable with respect to Borel reducibility.

同胚群作用的分类复杂性

Classification complexity of homeomorphism group actions

本文探讨紧致可度量化空间的同胚群被稠密非闭子群取代时轨道等价关系分类复杂性的变化，分析三类子群及两类空间的超空间作用，发现其复杂性与闭子群不同。

Countably compact group topologies on arbitrarily large free Abelian groups

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $κ$ such that $κ^ω=κ$, there exists a group topology on the free Abelian group of cardinality $κ$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

Higher dimensional obstructions for star reductions

A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $*$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induc…

自由群的自由III型作用的弹性

Elasticity of free type III actions of free groups

本研究证明保测度类的非顺从可树化III型等价关系可由任意生成元数的非阿贝尔自由群的自由作用诱导且对应Schreier图端不消失，借助已有III型刻画完成证明，凸显了III型与保测度情形的差异。