Bijective Quasi-Isometries of Amenable Groups

Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$κ$-to-one in a natural sense for some $κ>0$. For non-amenable graphs, all quasi-isometries are quasi-$κ$-to-one for any $κ>0$, while for amenable ones there exists at most one possible such $κ$. For an amenable graph $X$, we show that the set of possible $κ$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a giv…

Amenable groups and a geometric view on unitarisability

We investigate unitarisability of groups by looking at actions on the cone of positive invertible operators of a Hilbert space. This way, we give a geometric prove to a result by Gilles Pisier on the existence of some universal constants for unitarisable groups. By constructing barycenters for finite sets in a some class of geodesic metric spaces, we prove a fixed point theorem for actions of amenable groups by isometries. In particular, we show that extensions of unitarisable groups by amenable groups are unitarisable and virtual unitarisability implies unitarisability and estimate the corres…

Similarity problems, Folner sets and isometric representations of amenable semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).

Gauge-Invariant Uniqueness and Reductions of Ordered Groups

A reduction $φ$ of an ordered group $(G,P)$ to another ordered group is an order homomorphism which maps each interval $[1,p]$ bijectively onto $[1, φ(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and reduces to an amenable ordered group, then there is a gauge-invariant uniqueness theorem for $P$-graph algebras. We also consider the class of ordered groups which reduce to an amenable ordered group, and show this class contains all amenable ordered groups and is closed under direct products, free products, and hereditary subgroups.

Fixed points and amenability in non-positive curvature

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot …

类灯群的分析、几何与度量方面

Analytic, geometric and measured aspects of lamplighter-like groups

本论文从拟等距与度量角度研究类灯群，给出置换圈积的拟等距分类及刚性结果，并构造晕积的轨道等价耦合，改进等周轮廓估计。