On a class of finite upper half-planes
type: Item
instance of: scholarly article
title: On a class of finite upper half-planes
zbMATH Open document ID: 0876.11057
DOI: 10.1016/S0012-365X(97)89266-7
author: Mihai Caragiu
published in: Discrete Mathematics
publication date: 1997-07-07T00:00:00Z
review text: Inspired by the finite upper half plane graphs considered in [\textit{A. Terras}, Exp. Math. 5, 15-32 (1996; Zbl 0871.05044)]\ with their finite-field valued analogue of Poincaré's distance, the author produces a real-valued distance on the finite upper half-plane over the finite field with q elements and some new associated graphs \(V(q;k)\). This distance is shown to be a character sum. Basic results on numbers of points on a complete elliptic curve over a finite field or character sum estimates of Weil are required. Unlike the \((q+1)\)-regular finite upper half-plane graphs of the reviewer, these \(V(q;k)\) graphs may have no edges. But it seems that when renormalized, the nonempty \(V(q;k)\) graphs are `small perturbations of complete graphs'. The original finite upper half-plane graphs are of interest because of their connections with the finite Gelfand pair \((G,K)\), where \(G\) is the general linear group of \(2\times 2\) non-singular matrices with entries in a finite field \(F\), and \(K\) is analogous to the rotation group \(O(2)\). Moreover, they provide examples of Ramanujan graphs in the sense of Lubotzky, Phillips and Sarnak. [See \textit{A. Lubotzky}, Discrete groups, expanding graphs, and invariant measures. Birkhäuser (1994; Zbl 0826.22012)]. But no study of the spectra of the adjacency matrices of these new finite upper half-plane graphs seems to have been made. Generalizations of finite upper half-planes to higher dimensions are also considered by considering higher degree extensions of the base field. (Also submitted to MR).
reviewed by: Audrey A. Terras
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1019020
zbMATH Keywords: finite upper half plane graphs
zbMATH Keywords: distance
zbMATH Keywords: character sum
zbMATH Keywords: Ramanujan graphs
MaRDI profile type: Publication
cites work: Q3138326
cites work: Q4314388
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cites work: Lectures on the theory of algebraic functions of one variable
cites work: Q5690468
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full work available at URL: s0012-365x(97)89266-7
OpenAlex ID: W2061896879
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description: scientific article; zbMATH DE number 1019020