Uncountably many $2$-generated just-infinite branch pro-$2$ groups

Uncountably many permutation stable groups

In a 1937 paper B.H. Neumann constructed an uncountable family of $2$-generated groups. We prove that all of his groups are permutation stable by analyzing the structure of their invariant random subgroups.

Uncountably many quasi-isometry classes of groups of type $FP$

Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.

短层级双曲群 I：不可数多个粗中值结构

Short hierarchically hyperbolic groups I: uncountably many coarse median structures

本文证明五穿孔球面映射类群及某些右角Artin群具有不可数多个粗中值结构，通过发展短层级双曲群理论并利用拟态射构造拟直线实现。

The number of locally invariant orderings of a group

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.

Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups

C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.

Infinitely presented C(6)-groups are SQ-universal

We prove that infinitely presented classical $C(6)$ small cancellation groups are SQ-universal. We extend the result to graphical $Gr_*(6)$-groups over free products. For every $p\in\mathbb{N}$, we construct uncountably many pairwise non-quasi-isometric groups that admit classical $C(p)$-presentations but no graphical $Gr'(\frac{1}{6})$-presentations.

Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

几何无限作用的两种刻画

Two Characterizations of Geometrically Infinite Actions on Gromov Hyperbolic Spaces

本文通过逃逸测地线和不可数非锥形极限点的存在性，为几何无限作用提供了两种新刻画，扩展了Bonahon、Bishop及Kapovich-Liu在Kleinian群和负曲率流形中的定理。

Dehn function and asymptotic cones of Abels' group

We prove that Abels' group over an arbitrary nondiscrete locally compact field has a quadratic Dehn function. As applications, we exhibit connected Lie groups and polycyclic groups whose asymptotic cones have uncountable abelian fundamental group. We also obtain, from the case of finite characteristic, uncountably many non-quasi-isometric finitely generated solvable groups, as well as peculiar examples of fundamental groups of asymptotic cones.

On the number of subsemigroups of direct products involving the free monogenic semigroup

The direct product $\mathbb{N}\times\mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise non-isomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise non-isomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise non-isomorphic subdirect products if and only if every element of $S$ has a relative left- or right identity element.

Orbit inequivalent actions for groups containing a copy of $\Bbb F_2$

We prove that if a countable group $Γ$ contains a copy of $\Bbb F_2$, then it admits uncountably many non orbit equivalent actions.

A class of II_1 factors with many non conjugate Cartan subalgebras

We construct a class of II_1 factors M that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of M is complete analytic, in particular non Borel. We also construct a II_1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II_1 factors are all isomorphic.

Orbit Equivalence and actions of F_n

In this paper we show that there are "E_0 many" orbit inequivalent free actions of the free groups F_n, $2\leq n\leq\infty$, by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

Hyperbolic groups with almost finitely presented subgroups

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.