The Structure and Spectrum of Heisenberg Odometers
نویسندگان
چکیده
Odometer actions of discrete, finitely generated and residually finite groups G have been defined by Cortez and Petite. In this paper we focus on the case where G is the discrete Heisenberg group. We prove a structure theorem for finite index subgroups of the Heisenberg group based on their geometry when they are considered as subsets of Z3. We use this structure theorem to provide a classification of Heisenberg odometers and we construct examples of each class. In order to construct some of the examples we also provide necessary and sufficient conditions for a Zd odometer to be a product odometer as defined by Cortez. It follows from work of Mackey that all such actions have discrete spectrum. Here we provide a different proof of this fact for general G odometers which allows us to identify explicitly those representations of the Heisenberg group which appear in the spectral decomposition of a given Heisenberg odometer.
منابع مشابه
Linear Recursive Odometers and Beta-expansions
The aim of this paper is to study the connection between different properties related to β-expansions. In particular, the relation between two conditions, both ensuring pure discrete spectrum of the odometer, is analysed. The first one is the so-called Hypothesis B for the G-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Piso...
متن کاملحالت RVB بهینه در شبکه مربعی: حالت پایه و برانگیخته
We consider RVB state as a variational estimate for the ground state of Heisenberg antiferromagnet in square lattice. We present numerical calculation of energy, spin-spin correlation function and spin excitation spectrum. We show, that the quantum flactuations reduce of magnetization respect to Neel order. Our results are in good agreement with other methods such as spin-wave calculation a...
متن کاملTranslation invariant surfaces in the 3-dimensional Heisenberg group
In this paper, we study translation invariant surfaces in the 3-dimensional Heisenberg group $rm Nil_3$. In particular, we completely classify translation invariant surfaces in $rm Nil_3$ whose position vector $x$ satisfies the equation $Delta x = Ax$, where $Delta$ is the Laplacian operator of the surface and $A$ is a $3 times 3$-real matrix.
متن کاملB-FOCAL CURVES OF BIHARMONIC B-GENERAL HELICES IN Heis
In this paper, we study B-focal curves of biharmonic B -general helices according to Bishop frame in the Heisenberg group Heis Finally, we characterize the B-focal curves of biharmonic B- general helices in terms of Bishop frame in the Heisenberg group Heis
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012