Groups of hierarchomorphisms of trees and related Hilbert spaces

The subgroup $PSL(2,R)$ is spherical in the group of diffeomorphisms of the circle

We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.

On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits trees

Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We present a combinatorial description of the space of double cosets of $Hie(T_n)$ with respect to $Aut(T_n)$ and construct a 'ne…

On the group of spheromorphisms of the homogeneous non-locally finite tree

Consider a tree $\mathbb T$, all whose vertices have countable valence; its boundary is the Baire space $\mathbb{B} \simeq\mathbb{N}^{\mathbb N}$; continued fractions expansions identify the set of irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ with $\mathbb B$. Removing $k$ edges from $\mathbb T$ we get a forest consisting of copies of $\mathbb T$. A spheromorphism (or hierarchomorphism) of $\mathbb T$ is an isomorphisms of two such subforests considered as a transformation of $\mathbb T$ or of $\mathbb B$. Denote the group of all spheromorphisms by $\mathrm{Hier}({\mathbb T})$. We a sho…

树自同构群到庞特里亚金空间的不可约表示

Irreducible representations of tree automorphism groups into Pontryagin spaces

研究树自同构群在有限索引连续强非退化半双线性形式上的连续不可约表示，证明索引大于1时无更多表示，完成分类。

Thompson Field Theory

We introduce Thompson field theory, a class of toy models of conformal field theory in which Thompson's group T takes the role of a discrete analogue of the chiral conformal group. T and the related group F are discrete transformations of dyadic partitions of the circle and the unit interval, respectively. When vectors or tensors are associated with partitions, one can construct a direct limit Hilbert space, here called the semicontinuous limit, and F and T have unitary representations on this space. We give an abstract description of these representations following the work of Jones. We also …