The finite upper half space and related hypergraphs
type: Item
instance of: scholarly article
title: The finite upper half space and related hypergraphs
zbMATH Open document ID: 0989.11067
author: María G. Martínez
published in: Journal of Number Theory
publication date: 2001-01-01T00:00:00Z
review text: The finite upper half plane and the Ramanujan graphs attached to it have as their origin an attempt to find a simple model of the Poincaré upper half plane. In the paper under review a higher rank analogue is found that may be viewed as providing a finite model for the symmetric space of the \(n\times n\) general linear group. See the reviewer's book [\textit{A. A. Terras}, Fourier analysis on finite groups and applications, Cambridge Univ. Press (1999; Zbl 0928.43001)] for a discussion of the finite upper half plane. Other references are: \textit{Wen-Ching Winnie Li} [Eigenvalues of Ramanujan graphs, in IMA Vol. Math. Appl. 109, 387-403 (1999; Zbl 0981.11041)], \textit{A. Lubotzky} [ Discrete groups, expanding graphs and invariant measures, Birkhäuser (1994; Zbl 0826.22012)], and \textit{A. Valette} [Sémin. Bourbaki 1996/97, Exp. No. 829, Astérisque 245, 247-276 (1997; Zbl 0929.05042)]. Let \(\delta\) be a non-square in a finite field with \(q\) elements, denoted \(\mathbb{F}_{q}\) , where \(q\) is an odd prime power. The finite upper half plane \(H_{q}\) is the set of all \(z=x+y\sqrt{\delta}\), such that \(x,y\) \(\in\) \(\mathbb{F}_{q}\) and \(y\) is non-zero. A finite analogue of the Poincaré non-Euclidean distance is defined for \(z,w\) \(\in\) \(H_{q}\) by \[ d(z,w)=\frac{N(z-w)} {\text{Im }z\text{ Im }w},\quad \overline{z}=x-y\sqrt{\delta},\quad Nz=z\overline{z},\quad \text{Im }z=y. \] This distance can be used to define the \textit{finite upper half plane graph} \(X_{q}(\delta,a)\) whose vertices are the elements of \(H_{q}\). Then put an edge between vertices \(z\) and \(w\) if \(d(z,w)=a\). The graph \(X_{3}(2,1)\) is an octahedron, for example. The finite group \(G=\text{GL}(2,\mathbb{F}_{q})\) of all \(2\times 2\) non-singular matrices with entries in \(\mathbb{F}_{q}\) acts on \(z \) in \(H_{q}\) by fractional linear transformation; i.e \(\frac{az+b}{cz+d}. \) Thus one can identify \(H_{q}\) with \(G/K\), where \(K\) is the subgroup of \(G\) fixing \(\…
Mathematics Subject Classification ID: 11T60
Mathematics Subject Classification ID: 05C65
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1547369
zbMATH Keywords: finite upper half space
zbMATH Keywords: Ramanujan hypergraph
zbMATH Keywords: general linear group over a finite field
zbMATH Keywords: finite upper half plane graph
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: jnth.2000.2524
OpenAlex ID: W2091948927
cites work: Finite upper half planes over finite fields
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cites work: Discrete groups, expanding graphs and invariant measures. Appendix by Jonathan D. Rogawski
cites work: Ramanujan graphs
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cites work: Q3994578
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cites work: Radon transform on finite symmetric spaces
DOI: 10.1006/JNTH.2000.2524
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description: scientific article; zbMATH DE number 1547369